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Making Logarithms Accessible – Operational and Structural Basic Models for Logarithms

Zugänge zum Logarithmus – Operationale und strukturelle Grundvorstellungen zum Logarithmus

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Abstract

Logarithms have a reputation for being difficult and inaccessible. As an analysis of their historical, mathematical and educational background suggests, this problem might be due to the way in which logarithms are interpreted and explained in textbooks: as the inverse of exponents. If this conclusion is right, additional interpretations of logarithms are required.

By combining the theoretical construct of ‘Grundvorstellungen’ (translated as ‘basic models’) and the distinction between operational and structural conceptions, I identify and elaborate four interpretations of logarithms: (i) the basic model of ‘multiplicative measuring’, (ii) the basic model of ‘counting the number of digits’, (iii) the basic model of ‘decreasing the hierarchy level’, and (iv) the basic model of ‘inverse exponent’. Three models (i–iii) reflect operational conceptions and interpret logarithms in contexts familiar to students. In combination with (iv), a structural basic model, this paper argues on a theoretical level that they could help to make logarithms accessible and understandable to students. Following the tradition of ‘Stoffdidaktik’ (‘subject-matter didactics’), the study thus aims to unpack some of the content knowledge required for the teaching of logarithms.

Zusammenfassung

Der Logarithmus gilt als schwierig und unverständlich. Wie eine historische, mathematische und didaktische Sachanalyse zeigt, könnte dieses Problem darauf zurückgehen, dass der Logarithmus in Schulbüchern primär als inverser Exponent eingeführt und interpretiert wird. Wenn diese Diagnose zutrifft, sind weitere Zugänge zum Logarithmus gefragt.

Der vorliegende Beitrag entwickelt und diskutiert – unter Bezugnahme auf das theoretische Konstrukt der Grundvorstellungen sowie auf die Unterscheidung zwischen operationalen und strukturellen Auffassungen mathematischer Begriffe – vier Grundvorstellungen zum Logarithmus: (i) die Grundvorstellung des ‚multiplikativen Einpassens‘, (ii) die Grundvorstellung des ‚Bestimmens der Stellenzahl‘, (iii) die Grundvorstellung des ‚Herabsetzens der Hierarchiestufe‘, sowie (iv) die Grundvorstellung des ‚inversen Exponenten‘. Drei Interpretationen (i bis iii) deuten den Logarithmus operational und darüber hinaus in einem Erfahrungsbereich, der den Lernenden vertraut ist. Zusammen mit (iv), einer strukturellen Grundvorstellung, könnten sie – so wird hier aus theoretischer Sicht argumentiert – den Logarithmus für Lernende zugänglicher und verständlicher machen. In der Tradition der Stoffdidaktik stehend, bereitet dieser Beitrag mathematisches Professionswissen für das Unterrichten des Logarithmus auf.

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Notes

  1. For a comprehensive overview of ‘Grundvorstellungen’ identified so far for lower secondary concepts, see Jordan (2006, pp. 148–154).

  2. In this paper, the term “context” is not limited to real-life settings, but is considered more broadly in the sense of meaningful settings where particular conceptualizations of a mathematical concept can be activated (e.g. Leuders et al. 2011). Moreover, the term “familiar” is used to refer to students’ individual experiences and underlying activities (e.g. comparing, counting, etc.). In the German tradition, the corresponding construct is ‘subjektiver Erfahrungsbereich’ (Bauersfeld 1983; vom Hofe 1998), or ‘subjective experiential domain.’

  3. As early as 1850, a German textbook proposes “Grundvorstellungen zum Logarithmus” (basic models for logarithms), with the aim of explaining logarithms in a more “comprehensible and thorough” manner (Matzka 1850). However, he interpreted them as “[…] representatives, chargés d’affaires, authorised agents […]” (ibid., p. 10, translated by C. W.), so one can hardly consider these as basic models in the modern sense.

  4. Contrary to the English-speaking tradition where the obelus sign “\(\div\)” is used as a division symbol, German-speaking countries use the colon “:” to indicate the division operation (also used for ratios).

  5. To do and write down division problems like \(b\div a\) with long division, English-speaking countries use the symbol . Read as “\(a\) is divided into \(b\),” the symbol conveys the quotative interpretation of division (cf. Sect. 2.1.4). In contrast, this symbol is not known in German-speaking countries where the division symbol \(b:a\) is also used for writing down long divisions.

  6. If \(b<a\), the equivalence \(\log_{a}b=1/\log_{b}a\) can be used.

  7. This interpretation can be further generalised to arguments written in any non-decimal base, by changing the base of the logarithm. For example, the base 2 logarithm of a number \(b\) finds the number of digits minus 1 in the binary expression of \(b\).

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Weber, C. Making Logarithms Accessible – Operational and Structural Basic Models for Logarithms. J Math Didakt 37 (Suppl 1), 69–98 (2016). https://doi.org/10.1007/s13138-016-0104-6

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