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After the Hindu-Arabic decimal positional system was introduced in Europe, throughout many centuries textbooks on elementary arithmetic, intended for beginners, had a more or less fixed organization of content, usually starting with chapters on numeration. These chapters, as a rule, contained one or more examples of large integers the purpose of which was simply to be named (read out loud), sometimes also vice versa. This tradition apparently began with the two first texts that significantly contributed to the spread of the decimal system in Europe--the Latin translations of al-Khwarizmi's treatise on decimal arithmetic, and Leonardo's Liber Abaci, containing examples of reading a 16-digit and a 15-digit number respectively. Throughout the centuries, the order of magnitude of these introductory numbers increased, in general up to some 30 digits, but in some cases to over 60 digits. In this paper we examine the development and extent of this characteristic of introductory arithmetic textbooks from the period 13th-19th century, and the conditions which lead to this, now extinct, practice. [ABSTRACT FROM AUTHOR]

This paper explores written calculation methods for division used by pupils in England (n = 276) and the Netherlands (n = 259) at two points in the same school year. Informal strategies are analysed and progression identified towards more structured procedures that result from different teaching approaches. Comparison of the methods used by year 5 (Group 6) pupils in the two countries shows greater success in the Dutch approach, which is based on careful progression from informal strategies to more structured and efficient procedures. This success is particularly notable for the girls in the sample. For the English pupils, whose written solutions largely involved the traditional algorithm, the discontinuity between the formal computation procedure and informal solution strategies presents difficulties. [ABSTRACT FROM AUTHOR]