Mathematics and Literature: Perspectives for interdisciplinary classroom pedagogy

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1 Mathematics and Literature: Perspectives for interdisciplinary classroom pedagogy Bharath Sriraman Dept. of Mathematics The University of Montana Astrid Beckmann University of Education Schwäbisch Gmünd Abstract: Mathematics and literature have an ancient affinity as seen in the writings of the Greek philosophers, medieval theologians and natural philosophers of the post Renaissance period. Many of these writings indicate a pragmatic style of doing mathematics as opposed to the essentialist nature conveyed by school curricula today. Given the modern Renaissance of sorts as seen in the explosion of expository literature in mathematics/science, we explore the use of literature in the teaching and learning of mathematics. In particular, we explore (1) the use of the genre of modern mathematics fiction such as Flatland and Flatterland for the purpose of introducing advanced mathematical ideas and to allow the exploration of philosophical questions on the nature of mathematics; and (2) The use of literary texts which contain mathematics in it in conjunction with the use of a special software called MATEX which connects short literary texts and mathematical learning. 1. Use of Mathematics Fiction (American Context) Flatland (Abbot, 1984) a book of mathematical fiction is the unusual marriage of literature and mathematics. Sriraman (2003) used this book as a didactic tool to provide the ideal scaffolding for year old students to critically examine societal norms and biases in addition to exposing students to some very advanced mathematical ideas such as dimension. It also created the perfect setting to expose students to non-euclidean geometries such as the Minkowskian space-time geometry and Fractal geometry. One purpose of introducing non-euclidean geometries was to expose students to relatively modern ideas that were instrumental in the subsequent foundational problems that plagued mathematics in the early part of the 20 th century. The hope was this exposure to non-intuitive mathematics would lead students to form a basis for an ontological and epistemological standpoint. A follow up teaching experiment (Sriraman, 2004a, 2004b) consisted of reading the first five chapters of Flatterland, a sequel to Flatland, in which Stewart (2001) brilliantly makes numerous non-euclidean geometries accessible to the lay person in addition to introducing ideas such as encryption on the Internet, the taxi-cab metric, and fractal geometry. These readings further exposed students to the non-intuitive aspect of mathematics. For instance Stewart (2001) describes the non-intuitive possibility of fitting a cube of side length 1.06 into a unit cube, which some students did not accept as being possible in spite of the sound mathematical argument in the book. The origins of this mathematical problem of fitting a larger cube into a smaller one was a wager made and won by Prince Rupert in the late 17 th century (Jerrard & Wetzel, 2004) about this bizarre possibility. The non-intuitive aspect of mathematics brought alive by Flatterland led students in Sriraman s studies to take an ontological position on the nature of mathematics. Eventually students also asked the epistemological question "How does one know truth in mathematics?" In other words students were questioning the nature of the truth of the non-intuitive possibilities of mathematics. The specific ontological question was whether the protagonist of Flatterland was discovering

2 geometries that were present a priori or were the different geometries a figment of imagination made real via the use of a virtual reality device in the book (Sriraman, 2004b). The specific epistemological question was whether one could believe in the truth of these new geometries? For some students that were interested in science fiction, these questions also coincided with questions about the nature of reality and truth raised by the film The Matrix in which reality as is was quite different from reality experienced through a virtual interface (Sriraman, 2004a, 2004b). So students were not only posing the analogous question for mathematics but were also willing to consider this question independent of the context of a specific intellectual discipline. The didactic goal here to let students formulate their own philosophical questions and try to answer them (Sriraman & Dahl, 2007, in press). The use of mathematics fiction served as a didactic tool to scaffold this process and also lends itself to differentiated learning and instruction in the form of special topics projects. Mathematics fiction serves as the ideal tool to both awaken the imagination and to explore deep philosophical questions. Use of Literary Texts (German Context) Based on the hermeneutic background, literary texts are a central subject matter in language education. In Germany there are no binding literature lists, but only recommended lists with a very extensive selection. In mathematics lessons the use of literary texts is not part of the normal curriculum, but there are many literary texts concerning mathematics. This allows for the possibility of choosing mathematical literature even in language lessons and vice-versa. This leads to an interdisciplinary challenge. A categorization of such texts and the possibilities for mathematics learning are presented. Category A: Mathematics in a small part of the whole (long) literary text Example (Goethe 2005) (From Faust: The witch s one-time-one) This must ken! From one make ten, And two let be, Make even three, Then rich you ll be. Skip oér the four! From five and six, The Witch s tricks, Make seven and eight, Tis finished straight; And nine is one, And ten is none, That is the witch s one-time-one! (Interpretation: The idea of the text could be interpreted by the magical square shown above)

3 Category B: Short literary texts mathematics as a main concern The typical mathematical text in this category is a poem or a parable. Example (Kästner 1978 with author s own translation): Compassion and Perspective or The View of a Tree Here, where I stand, here we are trees, incomparable big and wide the street and the spaces in between. My God, I feel so sorry for the small trees at the end of the street! (Interpretation is the figure shown here) Depending on the different categories of mathematical literature its integration into mathematical lessons may differ. A long text with an important mathematical part can be read in conjunction with language/literature lessons. Possibly the mathematical content can contribute to the understanding of the text. In most cases literature motivates special mathematical themes. The poem of the tree mentioned above could lead to the construction of the accompanying picture or in general to questions of descriptive geometry. Category C: The mathematics concerns the whole (long) literary text Typical texts in this category are, for example, the famous stories by Gardener and Stewart written for Scientific American or Pour la Science. Here the mathematical idea is presented in entertaining stories. We can find this kind of literature in childrens books (Moore, Bintz 2002). Examples: What comes in 2s, 3s, and 4s is a story about number concepts (Fraser 1993). A list of novels is annotated by Usnick et al (Usnick, McCarthy 1998; see also Martinie et al 2003 and Lipsey et al 2002). These stories broach mathematical topics like ratio of quantities, proportional relations, and similarity. These types of literary texts are also represented by the story Through the looking glass although Lewis Carrol was a mathematician and his story was motivated by mathematical ideas. We present two examples from category C. We look at initial literary texts with some kind of mathematical background and distinguish two kinds of books. The first type is characterized by the fact that the literary style is set against the backdrop of mathematical knowledge and interest. The second type is represented by books in which a problematic (affective) view of mathematics is presented in the literature. The second type is not presented here due to space constraints. The first example: The Rider on the White Horse written by the famous German author Theodor Storm (Storm 1917). The novel is the story of a dikemaster at the North-Frisian coast in the 18 th century. The young dikemaster fails as he tries to base the construction of dikes on scientific ideas. So the novel deals with mathematical and technical questions. The mathematics is used to illustrate the problem of individual isolation (Radbruch 1997).

4 The novel begins with the characterisation of the main person, Hauke, the son of the old dikemaster, who in winter sat in his room to measure and calculate. His son, the (future) dikemaster, usually sat by and watched his father perform calculations and asked critically the basis of the calculations. But his father, who did not know how to answer this, shook his head and said: That I cannot tell you; anyway it is so...if you want to know more, search for a book to-morrow in a box in our attic; someone with the name of Euclid has written it; that will tell you The next day the boy ran up the attic and soon found the book, and soon became absorbed in Euclid s Elements. When the old man saw that the boy had no sense for cows or sheep and scarcely noticed when the beans were in bloom, which is the joy of every marsh man... he sent his boy to the dike, where he had to cart from Easter to Martinmas. That will cure him of Euclid, he said to himself... And the boy carted; but his Euclid he was always with him in his pocket, and when the workmen ate their breakfast or lunch, he sat on his upturned wheelbarrow with the book in his hand. In autumn, when the tide rose higher and sometimes work had to be stopped, he did not go home with the others, but stayed and sat... and for hours watched the sombre waves of the North Sea beat higher and higher against the grass-grown scar of the dike.... After staring a long time, he would nod his head slowly and, without looking up, draw a curved line in the air, as if he could in this way give the dike a gentler slope. Our dikes aren t worth anything... they re no good father, said Hauke... The waterside is too steep, he said, íf it happened some day as it has happened before, we can drown here behind the dike too. This novel touches or stimulates the following aspects of mathematics: - area of triangle, parallelogram, and trapezium: calculation of the quantities of ground which is needed for a given profile of the dike. To formulate and answer the questions we fall back on historical works. One example are the sketches of Heistermann (17 th century), which suggest the following masses and discuss different possibilities for calculating the area. Figure 1 Measures of a dike, according to Heistermann 17 th century, F = feet, R = Rhute, old Frisian unit of length, Beckmann Calculation in different measures: calculation of the ground and costs in historical units (Here we can fall back on data in historical works as well, Beckmann 2003)

5 - Triangles, sentences of congruence, Here we answer the question: What did the dikemaster learn from Euclid? Example: determining the profile of the dike by knowing the height and the gradient of the dike (Beckmann 2003) Figure 2 determining the profile of the dike, using congruence notions The use of MATEX Software MATEX is a computer environment which connects literature and mathematical learning (MATEX: discover Mathematics in literary TEXts was developed by Beckmann). It is Internetbased(visit matex.net.tc) and contains different interfaces. There are three frames, one containing the poems/ the mathematical texts, the other containing questions and hints and the last containing programs for the documentation of ideas and results. The special purpose of the program is that students keep a learning diary while analysing a special text. The first task for the students is to choose a text in a frame and to think about it. The students have to approach the text aesthetically and have to use their imagination. While navigating the students get general questions/prompts or concrete hints. Stimulating questions like: Does the text describe an actual situation? Could you formulate it mathematically? Does the text suggest the past and the future? How? For example: Could you give an equation or draw a graph? Etc. Critical questions like: What part does the mathematics play in relation to the literary content of the text? Did the author think about a mathematical background? Does the mathematical description correspond to the content, and help to better understand the text or distract it? And so on Another central component of MATEX is that mathematics arising from mathematical themes touched by the poems. Algebraic, geometric, stochastic, and other concepts are presented independently of the texts. So the students can repeat and learn the connected mathematics and pick up some ideas for their interpretation of the poem. In conclusion, we have presented a very brief synopsis of the possibilities for mathematics pedagogy from the use of mathematics fiction and literature. Space constraints prevent us from presenting other examples. But our hope is that the community of teachers and learners realizes

6 the amazing mathematics potential in historical and contemporary literature which are situated in context and provoke the imagination of both teacher and learner. References Abbot, E. (1984). Flatland (Reprint of 1884 edition). Signet Classic Books. Beckmann, A. (2003): Interdisciplinary mathematics instruction. Partt. 3: Mathematics instruction in co-operation with the school-subject German. Faecheruebergreifender Mathematikunterricht. T. 3, Hildesheim, Berlin: Franzbecker Beckmann, A. (since 2000): MATEX an environment for mathematical learning through literature, Fraser; D. (1993): Maths plus literature:: why link the two? New Zealand Mathematics Magazine 30(2), 8-12 Goethe, J.-W. (2005): Faus (translation: G.M.Priest). Retrieved February 2005, Jerrard, R. P. & Wetzel, J. E. (2004). Prince Rupert s rectangles. The American Mathematical Monthly, 111 (1) Kästner, E. (1978): Compassion and perspective, Mitleid und perspective. In Das Erich Kästner Lesebuch. Zürich: Diogenes, 103 Lipsey, S. & Pasternack, B. (2003): Mathematics in literature. Proceedings of Int. Conference on the Humanistic Renaissance in Math. Education, Italy. Martinie, S., & Bay-Williams, J. (2003): Using literature to engage Students in proportional reasoning, Mathematics Teaching in the Middle School, 9(3), Moore, S.D., Bintz, W. (2002): Teaching geometry and measurement through literature, Mathematics Teaching in the Middle School, 8/2, Radbruch, K. (1997): Mathematical tracks in literature, Mathematische Spuren in der Literatur, Darmstadt: Wiss. Buchgesellschaft Sriraman, B. (2003). Mathematics and Literature: Synonyms, Antonyms or the Perfect Amalgam. The Australian Mathematics Teacher, 59 (4), Sriraman, B. (2004a). Mathematics and Literature (the sequel): Imagination as a pathway to advanced mathematical ideas and philosophy. The Australian Mathematics Teacher. 60 (1), Sriraman,B. (2004b). Re-creating the Renaissance. In M. Anaya, C. Michelsen (Eds),

7 Proceedings of TSG21: Relations between mathematics and others subjects of art and science: The 10th International Congress of Mathematics Education, Denmark, Sriraman, B. & Dahl, B. (2007). On bringing interdisciplinary Ideas to Gifted Education.To appear in L.V. Shavinina (Ed). The International Handbook of Giftedness. Springer Science Stewart, I. (2001). Flatterland. Perseus Publishing. Storm, Th. (1917): The Rider on the White Horse: The Harvard Classics Shelf of Fiction, Retrieved February 2005, Usnick, V., & McCarthy, J. (1998): Turning adolescents onto mathematics through literature, Middle School Journal, 29(4), 50-54