Abstract
Starting with the question whether there is a connection between the mathematical capabilities of a person and his or her mother tongue, we introduce a new modeling approach to quantitatively compare natural languages with mathematical language. The question arises from educational assessment studies that indicate such a relation. Texts written in natural languages can be deconstructed into a dependence graph, in simple cases a dependence tree. The same kind of deconstruction is also possible for mathematical texts. This gives an idea of how to quantitatively compare mathematical and natural language. To that end, we develop algorithms to define the distance between graphs. In this paper, we restrict the structure to trees. In order to measure the distance between trees, we use algorithms based on previous work measuring the distance of neurons using the constrained tree edit distance. Once a distance matrix has been computed, this matrix can be used to perform a cluster analysis.
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Bibliography
Bisang, W. (2015). Hidden Complexity - The Neglected Side of Complexity and its Implications. Linguistics Vanguard, 1(1), 177–187.
Brückner, S., Förster, M., Zlatkin-Troitschanskaia, O., & Walstad, W. B. (2015). Effects of prior economic education, native language, and gender on economic knowledge of firstyear students in higher education. A comparative study between Germany and the USA. Studies in Higher Education, 40(3), 437–453. https://doi.org/10.1080/03075079.2015.1004235
Cauchy, A.-L. (1821). Cours d’Analyse de l’Ecole royale polytechnique. I.re Partie, L’Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi.
CoreNLP (2017). Stanford CoreNLP oreNLP CoreNLP -07. du Ro. Retrieved from https://stanfordnlp.github.io/CoreNLP/. Accessed: June 10 2017.
Euler, L. (1748). Introductio in analysin infinitorum. Opera Omnia, Serie 1, Vol 8.
Euler, L. (1770). Vollständige Anleitung zur Algebra, Bd. 1. Kaiserliche Akademie der Wissenschaften, St. Petersburg, 1770.
Hamming, R. W. (1950). Error detecting and error correcting codes. Bell Systems Technical Journal, 26, 147–160.
Heumann, H., & Wittum G. (2009). The tree-edit-distance, a measure for quantifying neuronal morphology. Neuroinformatics 7(3), 179–190.
Kilpeläinen, P., & Mannila, H. (1991). The tree inclusion problem. In Proc. Internat. Joint Conf. on the Theory and Practice of Software Development, Volume 1, (pp. 202–214).
Lagrange, J. L. (1797). Théorie Des Fonctions Analytiques, Contenant Les Principes Du Calcul Différentiel, Dégagés De Toute Considération D’Infiniment Petits ou d’Évanouissans, De limites Ou de Fluxions, Et Réduits A L’Analyse Algébrique Des Quantités Finies. Paris: Imprimerie de la République, Prairial an V.
Leibniz, G. W. (1684). Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus. Acta Eruditorum Lipsiae, 1684.
Levenshtein, V. I. (1966). Binary codes capable of correcting insertions and reversals. Soviet Physics Doklady, 10(8), 707–710.
NLTK (2017). Natural Language Toolkit. Retrieved from http://www.nltk.org/. Accessed: June 10 2017.
Selkow, S. (1977). The tree-to-tree editing problem. Information Processing Letters, (6)6, 184–186.
SpaCy (2017). https://spacy.io/. Accessed: June 10 2017.
Tai, K. (1979). The tree-to-tree correction problem. Journal of the Association for Computing Machinery, 26(3), 422–433.
Wagner, R., & Fischer, M. (1974). The string-to-string correction problem. Journal of the Association for Computing Machinery, 12(1), 168–173.
Walstad, W. B., Watts, M., & Rebeck, K. (2007). Test of understanding in college economics: Examiner’s manual (4th ed.). New York, NY: National Council on Economic Education.
Weierstraß, K. (1878). Einleitung in die Theorien der Analytischen Funktionen. Vorlesung, gehalten in Berlin 1878. Mitschr. von Adolf Hurwitz.
Wittum, G. (1982). Diplomarbeit, Mathematik, Universität Karlsruhe.
Zhang, K., Statman, R., & Shasha, D. (1992). On the editing distance between unordered labeled trees. Information Processing Letters, 42, 133–139.
Zhang, K. (1996). A constrained edit distance between unordered labeled trees. Algorithmica, 15, 205–222.
Zlatkin-Troitschanskaia, O., Brückner, S., Schmidt, S., & Förster, M. (2016). Messung ökonomischen Fachwissens bei Studierenden in Deutschland und den USA – Eine mehrebenenanalytische Betrachtung der hochschulinstitutionellen und individuellen Einfl ussfaktoren. Unterrichtswissenschaft, 44(1), 73–88. https://doi.org/10.3262/uw1601073
Zlatkin-Troitschanskaia, O., Förster, M., Brückner, S., & Happ, R. (2014). Insights from a German assessment of business and economics competence. In H. Coates (Ed.), Higher Education Learning Outcomes Assessment: International Perspectives (pp. 175–197). Frankfurt am Main: Lang. http://dx.doi.org/10.3726/978-3-653-04632-8
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Wittum, G., Jabs, R., Hoffer, M., Nägel, A., Bisang, W., Zlatkin-Troitschanskaia, O. (2018). A Concept for Quantitative Comparison of Mathematical and Natural Language and its possible Effect on Learning. In: Zlatkin-Troitschanskaia, O., Wittum, G., Dengel, A. (eds) Positive Learning in the Age of Information. Springer VS, Wiesbaden. https://doi.org/10.1007/978-3-658-19567-0_8
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