# Algorithms

**DOI:**https://doi.org/10.1007/978-3-030-15789-0_8

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## Definition

The word algorithm probably comes from a transliterated version of the name al-Khwarizmi (c. 825 CE), the Arabic mathematician who described how to solve equations in his publication *al-jabr w’al-muqabala*. An algorithm comprises a step-by-step set of instructions in logical order that enable a specific task to be accomplished. Due to its nature, it can be programmed into a computer, although some problems may not be computable or solvable by an algorithm. In his famous paper, Turing (1936) showed, among other things, that Hilbert’s Entscheidungsproblem can have no solution. He did this by proving “that there can be no general process for determining whether a given formula *U* of the functional calculus *K* is provable, i.e., that there can be no machine which, supplied with any one *U* of these formulae, will eventually say whether *U* is provable” (1936, p. 259).

## Keywords

Algorithm Computing History of mathematics Instrumental understanding Representation## References

- Duval R (2006) A cognitive analysis of problems of comprehension in a learning of mathematics. Educ Stud Math 61:103–131CrossRefGoogle Scholar
- Godfrey D, Thomas MOJ (2008) Student perspectives on equation: the transition from school to university. Math Educ Res J 20(2):71–92CrossRefGoogle Scholar
- Graham AT, Pfannkuch M, Thomas MOJ (2009) Versatile thinking and the learning of statistical concepts. ZDM Int J Math Educ 45(2):681–695CrossRefGoogle Scholar
- Gray EM, Tall DO (1994) Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. J Res Math Educ 26(2):115–141Google Scholar
- Hiebert J, Lefevre P (1986) Coneceptual and procedural knowledge in mathematics: an introductory analysis. In: Hiebert J (ed) Conceptual and procedural knowledge: the case of mathematics. Erlbaum, Hillsdale, pp 1–27Google Scholar
- Khoussainov B, Khoussainova N (2012) Lectures on discrete mathematics for computer science. World Scientific, SingaporeCrossRefGoogle Scholar
- Skemp RR (1976) Relational understanding and instrumental understanding. Math Teach 77:20–26Google Scholar
- Tall DO (1992) The transition to advanced mathematical thinking; functions, limits, infinity, and proof. In: Grouws DA (ed) Handbook of research on mathematics teaching and learning. Macmillan, New York, pp 495–511Google Scholar
- Thomas MOJ (2008) Developing versatility in mathematical thinking. Mediterr J Res Math Educ 7(2):67–87Google Scholar
- Turing AM (1936) On computable numbers, with an application to the Entscheidungsproblem. Proc Lond Math Soc 42(2):230–265Google Scholar
- Williams SR (1991) Models of limit held by college calculus students. J Res Math Educ 22:237–251CrossRefGoogle Scholar