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Making Logarithms Accessible — Operational and Structural Basic Models for Logarithms

July 2016
Journal für Mathematik-Didaktik 37(1):69–98

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University of Teacher Education Lucerne

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.

Role of the basic model of 'counting the number of digits' as intermediary between two contexts of logarithms

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Role of the basic model of 'decreasing the hierarchy level' as intermediary between two

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This is an Original Manuscript of an article published by Springer in Journal für

Mathematik-Didaktik, 2016, Vol. 37, Issue 1 Supplement, 69–98, available

online: http://link.springer.com/article/10.1007/s13138-016-0104-6.

Making Logarithms Accessible — Operational

and Structural Basic Models for Logarithms

Christof Weber

Pädagogische Hochschule Nordwestschweiz (School of Education, University of

Applied Sciences Northwestern Switzerland), Benzburweg 30, 4410 Liestal,

Switzerland

Tel. +41 61 925 77 63

Fax +41 61 925 77 66

E-mail: christof.weber@fhnw.ch

URL: http://www.fhnw.ch/people/christof-weber/

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.

Keywords logarithm · concept formation · subject-matter didactics

· Grundvorstellungen · basic models · operational-structural · content knowledge

for teaching · upper secondary school

Operationale und strukturelle Grundvorstellungen zum Logarithmus

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.

Schlüsselwörter Logarithmus · Begriffsbildung · Stoffdidaktik

· Grundvorstellungen · Operational–strukturell · Unterrichtsspezifisches

mathematisches Professionswissen · Sekundarstufe 2

Mathematics Subject Classification (2010): C34 · D44 · D74 · H24

Introduction

In their study on students’ understanding of pH values in chemistry, Watters and

Watters quote the following comment by a student: “I don’t know what the log

actually is, I only know where the button is on my calculator” (Watters and

Watters 2006, p. 280). Such comments are not uncommon — many students do

not understand what logarithms ‘actually are’ and struggle with the concept

(Berezovski 2004; Espedal 2015; DePierro and Garafalo 2008; Kenney 2005;

Williams 2011). This is especially problematic considering the indispensability of

the concept of logarithms for understanding various scientific and mathematical

phenomena (e.g., pH values, decibel scales, financial interest rates, etc.).

Logarithms, then, leave some room for improvement with respect to making

accessible (Kirsch 1977), a guiding principle for the conception and planning of

mathematics lessons.

Why, then, are logarithms meaningless and difficult for many students? And what

mathematical interpretations could make the concept more accessible to students

so that they grasp the concept, experiencing it as meaningful and being able to

apply it to solve mathematical problems? Rooted in the German tradition of

Stoffdidaktik, or ‘subject-matter didactics’ (Sträßer 2014), this paper analyses the

concept of logarithms to understand how it can be made accessible to, and

meaningful for, students. In particular, it aims to make explicit the elements

required for the teaching of logarithms, i.e. to unpack some of the related

specialized content knowledge (Ball et al. 2008, p. 400).

One theoretical construct that describes how mathematical concepts can be made

comprehensible to students is that of Grundvorstellungen, or ‘basic models’

(Griesel 1971; vom Hofe 1998; vom Hofe et al. 2006; Oehl 1962). Focusing on

meaningful interpretations of concepts, this construct has enjoyed popularity in

the German-speaking countries since its inception. In international literature,

another theoretical construct on how accessibility can be achieved is more

established, namely that of operationality and structurality (Sfard 1991, 2008).

While accessibility plays an important role in the work of Sfard, she also

emphasises other quite different aspects. Both theoretical constructs have been

applied to concepts such as rational numbers, division or functions (e.g., Jordan

2006; Sfard 1991, 2008). To the best of my knowledge, they have never been

combined to identify the features of a concept that could make it more accessible

to students. This is what this paper aims to do with respect to logarithms.

The first section sets out the theoretical framework, introducing the theoretical

construct of basic models to non-German-language readers, and summarizing the

theoretical construct of operationality and structurality. The second section

presents a historical and mathematical analysis of logarithms as a concept. In

particular, it explains that logarithms and division are mathematical analogous,

and it presents a method by which logarithmic values can be calculated manually.

The third section is on educational background. It looks at the place of logarithms

in the secondary curriculum and in algebra textbooks, and provides an overview

of the body of literature on learners' difficulties and on various proposals as to

how these could be addressed. In particular, it gives some reasons why the

textbook definition of logarithms could be responsible for many difficulties and

mistakes. As will be shown in the forth section, the construct of operationality and

structurality can be used to diversify the construct of Grundvorstellungen: First,

four basic models for logarithms as numbers and operators will be proposed, then

their theoretical potential for the handling of logarithms and for explaining them

operationally or structurally will be analysed. The final section formulates the

main hypotheses, and outlines some future research for investigating them

empirically.

In short, this is an argumentative research paper: It analyses the subject matter of

logarithms from a semantic and an epistemological viewpoint, and then, by

combining two different theoretical frameworks, it proposes how logarithms could

be made accessible to learners. It generates no empirical findings but rather

hypotheses, underpinned with theoretical arguments. I thus present a more

thorough theoretical analysis of the subject than in my earlier study (Weber 2013).

1 Two Conceptualisations of Understanding

1.1 ‘Grundvorstellungen’ as Basic Models

The German mathematics educator Kirsch (1977) considers making accessible to

be an important guideline for teaching. Among other aspects, he suggests

including “the surroundings of mathematics” (ibid., p. 104). The theoretical

construct of ‘Grundvorstellungen’ follows this tradition and takes Kirsch’s

suggestion one step further: To make mathematics accessible to learners,

‘Grundvorstellungen’ place mathematical concepts in familiar contexts. The

following presents an outline of some central aspects of the theoretical construct

referred to in section 4, following the works of vom Hofe (1998), vom Hofe et al.

(2006) and Kleine et al. (2005). For a more comprehensive discussion, see vom

Hofe and Blum (2016, in this issue).

1.1.1 A theoretical construct and its use in German-speaking countries

There is a long tradition in German-speaking countries of seeking to ensure that

learning is experienced as something meaningful and not merely mechanical. In

the second half of the 20th century, the theoretical construct of

Grundvorstellungen — translated as ‘basic models’ (cf. 1.1.2) — became

established in German-speaking theories on teaching and learning mathematics. It

was first elaborated for arithmetic and algebra at primary school level, and then

expanded to lower and upper secondary mathematics.

Initially, the construct was mainly seen prescriptively, describing the standard

interpretations of mathematical concepts or theorems to be learned, like the

“sharing and measuring” interpretations for dividing (cf. 1.1.4). For generations,

many German-speaking mathematics educators sought such meaningful

interpretations to use as theoretical categories to develop lessons, write textbooks

and structure curricula.1

A different, more empirical view of interpretations of mathematical concepts was

introduced into the German-language discourse by vom Hofe (1998). Vom Hofe

studied Grundvorstellungen more from the perspective of students, taking into

account their actual interpretations and the conceptions guiding their problem

solving and argumentation. Through this descriptive broadening,

Grundvorstellungen “[…] can be used to describe relations between mathematical

contents and the phenomenon of the individual generation of concepts,

characterizing […] constitution of meaning of mathematical concepts based on

familiar contexts and experiences […]” (vom Hofe 1998, p. 320, italics in

original). A similar approach to conceptions guiding students’ problem solving

and argumentation was already being discussed in other countries in the form of

tacit models (Fischbein 1989), concept images (Tall and Vinner 1991), or use

meanings (Usiskin 1991).

1.1.2 ‘Grundvorstellungen’ as basic models

In order to use the theoretical construct ‘Grundvorstellungen’ in an English-

language context, it is first necessary to find an appropriate term for it. To

describe the prescriptive guidelines for structuring lessons (see above), it could be

translated as “basic ideas” (vom Hofe 1998, p. 319). If the focus is more on

individual meaning making, the term is sometimes translated as “individual

images” (vom Hofe 1998, p. 326) or “mental models” (vom Hofe et al. 2006, p.

142; Kleine et al. 2005, p. 229). Because the present paper will elaborate

normative interpretations of logarithms that are prototypical for working

mathematically, the term basic models is preferred here (cf. Selter et al. 2012).

This is in keeping with the English-language tradition where the interpretations of

division as sharing and measuring, for instance, are seen as “models” (Brown

1981, p. 23; Greer 1992, p. 276). As this example suggests, basic models are

standard interpretations of a mathematical concept (division) in the world of

experience, be it real-life experiences (sharing, comparing) or consolidated school

knowledge (subtraction). By having a certain explanatory power, they can serve

1 For a comprehensive overview of ‘Grundvorstellungen’ identified so far for lower secondary

concepts, see Jordan (2006, pp. 148–154).

students as a basis for meaningful argumentation and for solving tasks, going

beyond arbitrary interpretations of concepts.

1.1.3 Two types of basic models

To conceptualise elementary and advanced concept formation, vom Hofe

distinguishes between primary and secondary basic models, following Fischbein's

distinction between primary and secondary intuitions (1987). If a basic model

refers to real-life experience, vom Hofe speaks of a primary basic model. In a

sense intermediaries between real life and mathematics, they are used to support

mathematical reasoning that is related to reality (Blum 2002) or — in contrast —

the modelling of real-life problems (vom Hofe et al. 2006).

But many advanced concepts, including logarithms, are difficult to interpret in an

everyday context with realistic objects and manual actions. Nevertheless, to make

these kinds of concepts meaningful it is important to interpret them in familiar

contexts. Here it is useful to draw on the school context of mathematical objects

and actions: secondary basic models. Models of this type can be seen as

intermediaries between different modes of representations (e.g., vom Hofe and

Blum 2016, in this issue; Stölting 2008). The secondary basic models for

logarithms that will be identified in section 4, however, are slightly different.

They have less to do with mediating between different modes of representation

than with mediating between contexts of different levels of familiarity: Every

basic model for logarithms will link a non-familiar context of logarithms with

another context that students are likely to be more experienced in, for example

that of measurement (fig. 4–7).2

Finally, it should be noted that basic models, as interpretations, can never cover

all relevant aspects of a mathematical concept. They usually have less universal

validity than mathematical concepts, precisely because they refer to specific

contexts. Thus, in order to have a broad basis for reasoning and understanding, a

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.’

mathematical concept must be represented by a variety of different basic models:

Each aspect has to be experienced as a phenomenon in its own right.

1.1.4 Basic models for division, and a first research question

Because logarithms will later be shown to be analogous to division (cf. 2.1.2), and

because a variety of different basic models for division have been identified, the

theoretical considerations outlined above will be shortly illustrated.

Although the division of natural numbers can be seen as the sharing of objects

(fair-sharing or partitive basic model), division also occurs in other everyday

contexts. Another, no less important interpretation is the splitting up into groups

of equal size (measurement or quotative basic model), sometimes formulated by

asking how many times the divisor fits or goes into the dividend (fitting-into basic

model). While the fair-sharing model cannot be applied to fractions, the fitting-

into model can:

30 ÷ 1

2= 60

, because

goes into 30 as an addend sixty times.

The corresponding chain of argument can be represented as in figure 1: While the

two boxes represent different contexts of division, the arrows indicate how the

basic model mediates between the contexts (adapted from Prediger 2009, p. 170).

This example illustrates what is meant by the assertion that basic models have a

certain explanatory power and can serve as a basis for meaningful argumentation.

What is 30 divided by

1 2

30 – 1 2 – …– 1 2 =30 – 60 ⋅1 2 =0

∴!!30 ÷1 2 =60

Division as

(additive) fitting-into

How often does

1 2

(as an addend) fit into 30?

Fig. 1 Role of the basic model of ‘fitting-into’ as intermediary between two contexts of division

(adapted from Prediger 2009, p. 170)

In addition to these primary basic models for division, the repeated subtraction of

equal subtrahends and inverse multiplication are secondary basic models for

division. It should be noted that the interpretation of division as repeated

subtraction leads to the long division algorithm (Brown 1981; Fosnot and Dolk

2001; Greer 1992; Griesel 1971; Padberg and Benz 2011; Radatz et al. 1998).

As mentioned above, no basic models in the sense of ‘Grundvorstellungen’ have

thus far been identified for logarithms.3 This lack gives rise to the following

question:

(Q1) What are some different basic models for the concept of logarithms?

Specifically, which interpretations have the explanatory power to relate

logarithms to familiar mathematical contexts and consolidated experiences?

An arbitrary collection of basic models would not answer this question

satisfactorily. Instead, with regard to their accessibility, different “types” of basic

models are needed. This can be achieved through the construct of operationality

and structurality, which is briefly described below.

1.2 Operational and Structural Conceptions

A quite different perspective is afforded by process-object theories, which

conceptualise understanding as a combination of processes and objects. This is the

basic thinking behind a number of theories, including relational and instrumental

understanding (Skemp 1978), procepts (Gray and Tall 1994), the APOS theory

(Dubinsky 1991; Dubinsky et al. 2002), and operational versus structural

conceptions (Sfard 1991, 2008). As the latter perspective will be fruitful for the

elaboration of basic models for logarithms, the distinction between operational

and structural conceptions will be briefly outlined here, following Sfard (1991,

2008).

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.

1.2.1 Dual nature of mathematical concepts

According to Sfard, mathematical concepts can essentially be conceived in two

ways: operationally or structurally. These two conceptions are not mutually

exclusive, but complementary: Depending on how it is approached or acted upon,

a mathematical content can be thought of from an operational or structural

perspective. In other words, concepts such as fractions, functions, etc. always

embody both of these conceptions. The dual nature of mathematical concepts

offers some conceptual affordances: “[…] the structural approach generates

insight; the operational approach generates result” (Sfard 1991, p. 28).

The genesis of a mathematical concept often begins with a method or an

algorithm, a process. As an example, Sfard (1991) cites the evolution of the

concept of numbers. Whether from the division of natural numbers to positive

fractions, whether from the root extraction of real numbers to imaginary numbers:

it always began with a process. A notation such as

1 2

, for example, was

understood operationally first as division, and only later structurally when it was

read as a symbol for an object.

With reference to Piaget, Sfard (1991) outlines a model for learning in which

students begin by learning operational interpretations of mathematical concepts,

then progress towards their structural interpretation. In her view, new concepts

should not be taught only from a structural perspective, but teaching should take

care to also address complementary, operational aspects. The qualitatively crucial

step to understanding then consists in “realizing” several processes as a new

entity: In reification, a number of previously unconnected process-related aspects,

possibly in different representations, become united in a single object (Sfard 1991,

p. 19).

Sfard has been working on the operationalization of these terms for a long time

and recently began to broaden her originally cognitive view with a discursive

dimension (Sfard 2008). Thinking, and especially concept development, is now

seen as a matter of communication. Operationality and structurality thus become

properties of discourse on a concept: They no longer lie in its nature or

conception but are found instead in the language that is used to talk about and

explain a concept. Sfard therefore defines reification somewhat differently now:

“Reification is the act of replacing sentences about processes and actions with

propositions about states and objects” (ibid., p. 44). This means that the

accessibility of concepts could improve if teachers and students were engaged in a

discourse on processes and actions (Sfard 1991).

Sfard is not the only one to suggest that, when learning new concepts, operational

conceptions precede structural ones as consecutive steps to be passed through.

This is also the premise of Dubinsky and his colleagues in their research on

concept formation (Dubinsky 1991; Dubinsky and McDonald 2002). They note,

however, that concepts need not necessarily develop in this order. Gilmore and

Inglis (2008) give empirical evidence justifying this view.

1.2.2 Second research question

As the example of

1 2

illustrates, the distinction between operationality and

structurality is meaningful even for numbers and operators, and not only for

functions. To date, possible ways of realizing a mathematical concept as process

and as object have been illustrated not only for arithmetic expressions, but also for

algebraic expressions and equations (Dubinsky 1991; Gray and Tall 1994; Sfard

1991). For example, understanding the idea of squaring operationally means

describing it as a computation, and understanding it structurally means explaining

it as a “static relation between two magnitudes” (Sfard 1991, p. 6). It seems,

however, that the operational-structural construct has not yet been explicitly

applied to logarithms. In combination with (Q1) above, the second question is as

follows:

(Q2) Which of the identified basic models for logarithms enable a discourse on

processes and actions, and which enable a discourse on states and objects?

This paper will identify several basic models for logarithms that differ with

respect to their operational-structural character. They will be called operational

basic models and structural basic models. Before the two questions can be

answered, the concept of logarithms must first be analysed in its historical and

educational context.

2 Historical and Mathematical Analysis of

Logarithms

2.1 History of Logarithms

2.1.1 The invention of Napier and Bürgi

Logarithms were invented in the early 17th century by Napier or Bürgi, depending

on the source. Both were looking for a way to simplify astronomical calculations

of very large numbers. Though acting independently of one another, they both

proceeded in a similar manner, considering the relations between two

progressions: a geometric progression increasing by a constant factor, and an

arithmetic progression increasing by a constant addend. Napier employed a

geometric model of two correlated points moving continuously along two

different lines: One point moves on a (limited) segment at an instantaneous

velocity proportional to the remaining distance, while the other point moves at a

constant velocity on a ray (towards infinity). The distance of the first point from

the starting point thus grows geometrically, while the distance of the second point

grows arithmetically (for details, cf. Berezovski 2007, p. 23f.; Cajori 1913, p. 6f.;

Panagiotou 2011, p. 9; Smith and Confrey 1994, p. 347ff.).

On the basis of this model and after years of manual calculations, both Napier and

Bürgi came up with a pair of corresponding series sufficiently dense for practical

use in computation. These were published in 1614 (Napier) and 1620 (Bürgi)

(Cajori 1913; Panagiotou 2011; Smith and Confrey 1994). Napier’s tables in

particular went on to be widely distributed and were also variously simplified and

amplified. Shortly thereafter, in 1622, the first slide rule was invented by placing

two sliding logarithmic scales next to each other (Stoll 2006).

2.1.2 Logarithms as numbers that count divisions

The meaning of the term “logarithm” is not self-evident; and nor is the meaning of

its etymological translation: “ratio number.” Napier, who coined the term,

explained it as follows (Cajori 1913, p. 7): Two consecutive numbers of the

(geometric) progression

, .… have a constant ratio. If the (arithmetic)

progression 0, 1, 2, . . . is mapped onto this progression, the expressions of the

arithmetic progression indicate how often the 1 (of the geometric progression)

was multiplied by the ratio

. After first speaking of “artificial numbers,” he later

called the numbers of the arithmetic progression the “number of ratios,” or

“logarithm.” In other words, the “number that counts ratios” of 8 to base 2 is 3,

because 1 must be multiplied by 2 three times to yield 8. This makes clear the

mathematical analogy between logarithms and division (fig. 2): While the

division of 8 by 2 asks for the number of addends, the question behind the

logarithm of 8 to base 2 is the number of factors. In other words: Just as divisions

are repeated subtractions, logarithms are repeated divisions. This conception of

logarithms as numbers that count divisions was prevalent during the 17th century,

but became less important in later definitions of logarithms.

log28=?

←→#

1×!2×!×2

" #$ %$ =8

←→#

8÷2÷!÷2

" #$ %$ =1

8÷2=?

←→#

0+2+!+2

" #$ %$ =8

←→#

8−2−!−2

" #$ %$ =0

Fig. 2 Mathematical analogy between logarithms and division

2.1.3 Euler’s and other conceptualisations

After Descartes introduced the modern symbolism

for powers of numbers in

1637, it was recognised that logarithms could also be interpreted as exponents.

Euler was one of the first to use the exponential property as a definition (Cajori

1913, pp. 13, 46 f., 116):

Resuming the equation

ab=c

, […] we take the exponent

such that

the power

becomes equal to a given number

; in which case this

exponent

is said to be the logarithm of the number

. (Euler 1822, p.

63)

For Euler, the logarithm is a particular exponent: The logarithm of

to base

the exponent by which

must be raised to yield

. Rather than stating directly

what logarithms are, Euler proceeds indirectly by conceiving it in terms of its

inverse, raising to a power (with reference to the exponent). Today’s collections

of formulas follow this indirect definition when they define the logarithm by the

equivalence relation

logab=x:⇔ax=b

From the perspective of mathematics, this reading of logarithms simplifies the

deductive structure of the earlier conceptualisation, which is why it is widely

used. Using Sfard's terminology (cf. 1.2.1), it expresses a structural

conceptualisation, as it is a static relation that refers to another concept (see 4.4.2).

But, as will be shown later, it could be responsible for many problems faced by

students (cf. 3.4).

Further properties of logarithms were discovered after Euler that can also be used

as definitions and hence are definitional. Since the late 17th century, for example,

it has been known that the (natural) logarithm can be conceived as the area of a

hyperbolic segment or can be expanded in a series (Cajori 1913; Panagiotou

2011). Another reading was added by Cauchy in the 19th century: Logarithmic

functions are (apart from a factor) the only continuous solutions

of the

functional equation

Φ(x⋅y)=Φ(x)+Φ(y)

(

x>0

y>0

). Each of these

equivalent properties discovered in the past highlights a particular aspect of

logarithms and could serve as definition from a logical-deductive perspective.

2.1.4 Various notations reflect various aspects

Napier did not use any abbreviations for logarithms, but as early as 1624 the

abbreviation “Log.” was used (Kepler), and Euler wrote “L.” or “log” (1822, p.

63). Other forms of notation do not refer to the term “logarithm,” emphasising

instead the analogy to division (fig. 2). In 1778, Burja used the double fraction bar

for the logarithm of

to base

, and in 1834 Schellbach wrote it with a

tricolon, as

⁝

. These two German mathematicians and teachers would, then,

have written or

⁝

2=3

, respectively, instead of

log28=3

, explaining

that 8 must be divided by 2 three times to yield 1 (Cajori 1952, p. 112, p. 114).4

4 Contrary to the English-speaking tradition where the obelus sign “

” is used as a division

symbol, German-speaking countries use the colon “:” to indicate the division operation (also

used for ratios).

2.2 Algorithm for the Manual Calculation of Logarithms

As shown above, the logarithm

logab

indicates “how often” the base

contained as a factor in

, which can be rewritten as

alogab=1

. This is why, in

analogy to the standard division algorithm (“long division”), it is possible to

specify a logarithm algorithm that calculates logarithmic values decimal by

decimal (Goldberg 2006). One proceeds by repeated division rather than by

repeated subtraction, and instead of “bringing down” the next number and “adding

a zero” to the remainder (i.e. to multiply it by ten), it is raised to the power of ten

(the base of the decimal system).

The method thus essentially consists of the following steps (for an example of

taking the logarithm of 40 to base 2, see fig. 3):5

· Step 1: Calculate how many times the base goes into the argument as a factor,

and add this number as a new (decimal) place to the result. In the example, 2

goes into 40 five times as a factor without exceeding it, so the result begins

with the number 5.

· Step 2: Next, place the value of “base to the power of calculated position”

underneath the argument of the logarithm and divide the argument by this

value (instead of subtracting). Note the result as a remainder underneath. In the

example,

40 ÷25=1.25

· Step 3: Now, exponentiate (instead of multiplying) the remainder by ten and

note the result as the new argument:

1.2510 ≈9.31

· Step 4: If the new argument equals 1 (instead of zero), the process has come to

an end and the result is established. Otherwise, repeating steps 1 to 3 will result

in the next digits of the decimal expansion of the result (3, 2, …).

5 To do and write down division problems like

b÷a

with long division, English-speaking

countries use the symbol

a b

. Read as “

is divided into

,” the symbol conveys the

quotative interpretation of division (cf. 1.1.4). 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.

log240 = !! 5.32…!!!!!!!!!!!!

= 40 !!2 = ? ⎯ →⎯ 2)!40!!!!!!!!!!!!!

!!!!! ÷32 = 25

( )

1.25

!!!!! 1.2510 ≈9.31

!÷8 = 23

( )

1.16

1.1610 ≈4.57

: 4 = 22

( )

…!!!!!!!!!!!!!!!

Fig. 3 Logarithm algorithm, using the notation

⁝

for logarithms, introduced by Schellbach,

and the notation

a)!b

, referring to the long division symbol

a!b

used in English-speaking

countries to express long divisions in writing

To show the correctness of the algorithm, the entries in figure 3 may be

reformulated as a series of equations, as follows:

log240 =log2(32 ⋅1.25) =5+log2(1.25) =5+110 ⋅log2(1.2510)≈5+110 ⋅log2(9.31) ≈

≈5+110 ⋅log2(8 ⋅1.16) =5.3 +110 ⋅log2(1.16) =5.3 +1100 ⋅log2(1.1610 )≈…

Two points regarding the logarithm algorithm are worth noting here:

· In contrast to division, logarithms do not (left) distribute over addition:

loga(b

1+b2)≠logab

1+logab2

. Because of this, in step 1 of the algorithm, one

must consider all digits of the numbers

and

at once, rather than process

them digit by digit as in long division.

· If

and

are natural numbers (where

does not necessarily have to be less

than

), this algorithm allows one to manually calculate the value of

logab

This is because only long divisions (step 2) and

-fold multiplications (step 3)

need to be calculated in the given set of algorithmic rules.6

Being an algorithm that “generates result” (Sfard 1991, p. 28), this logarithm

algorithm is seen as an operational conceptualisation of logarithms (cf. 1.2.1).

6 If

b<a

, the equivalence

logab=1 logba

can be used.

3 Educational background of logarithms

This section looks at several aspects of the teaching and learning of logarithms.

First, the place of logarithms in school curricula and algebra textbooks will be

briefly outlined. This is followed by an overview of the research literature about

students’ difficulties and a review of alternative instruction techniques. The last

section looks at why the introduction of logarithms as indirect exponents is

problematic (cf. 2.1.3).

It should be noted that logarithms as numbers or operators must be distinguished

both mathematically and epistemologically from logarithms as functions (Smith

and Confrey 1994). Although both aspects of logarithms appear in upper

secondary level curricula, space precludes discussion of their functional aspect.

This paper is therefore focusing only on logarithms as numbers and operators.

3.1 Logarithms in Secondary Curricula and in School Textbooks

Ever since Euler, logarithms have been included in school curricula. They were

used to simplify complicated manual calculations and remained relevant in

schools for centuries until the introduction of pocket calculators, which have to a

large extent put an end to this tradition. In the German-speaking countries, for

example, the subject has been dropped entirely from curricula for lower achieving

school students. However, at upper secondary level, logarithms are still of some

importance. They usually follow powers (with non-natural exponents) and

exponential functions in grades 10–11 (i.e. age 16–17). Besides the objective of

manipulating expressions, logarithms are taught to solve specific equations,

particularly in the context of exponential growth and decay, and sometimes to

model mathematical and real-life problems. The properties and rules that

determine how logarithms behave are very important here (logarithmic laws,

change of base theorem).

With the broad circulation of the Elements of Algebra (originally published in

German in 1765), the indirect definition

logab=x:⇔ax=b

became the

standard. Even today, many high school textbooks explain logarithms with this

formal relation. As a brief (and unsystematic) examination of some modern

algebra textbooks in German and English suggests, they all use this equivalence

relation to introduce logarithms as inverse exponents (e.g., Deller et al. 2000;

Griesel et al. 2004; Hanrahan et al. 2004) or as inverse exponential functions (e.g.,

Neill and Quadling 2000; Murdock et al. 2010). Some of the textbooks examined,

in order to motivate students, do first introduce to a “real-life” phenomenon that

can be solved by logarithms (e.g., Hirsch et al.. 2008), while others interpret

points of a graph of exponential growth in “reverse order,” i.e. by starting with

some

-values and reading off their

-values (e.g., Holliday et al. 2005; Kuypers

et al. 1995). Even in those instances where a textbook presents such an

interpretation in the introductory section, it rarely makes use of it for

argumentation, for instance to make the logarithmic laws plausible (for an

exception, cf. Lergenmüller and Schmidt 2004). In short, one gets the impression

that the primary interpretation of logarithms conveyed by textbooks is the indirect

conception proposed by Euler.

3.2 Students’ Difficulties and Mistakes: an Overview of Literature

Mathematics education research literature on logarithms (in English and German)

identifies and analyses two different sorts of difficulties faced by students:

specific mistakes in manipulating logarithmic expressions, and more general

problems in understanding the meaning of the logarithmic concept.

3.2.1 Mistakes in manipulating logarithmic expressions

Studies that investigate the handling of logarithms by students generally present

the mistakes made as episodic observations, in a non-exhaustive fashion and

without any theoretical grounding. To date, there seems to be no “systematic

empirical research in this context” (van Dooren et al. 2008, p. 325). Thus all that

can be provided here is a list of some incorrect calculations mentioned in the

literature:

· The expression

log(x+y)

is rewritten as

log x+log y

(Kaur and Boey 1994;

Yen 1999),

log(x)+log( y)

x⋅y

(Berezovski 2004),

log(x)−log( y)

log(x) / log( y)

(Lee and Heyworth 1999),

log(xy)

xlog(y)

(Chua and

Wood 2005),

log(x) / log( y)

log(x/y)

(Kaur and Boey 1994) or as

x/y

(Chua and Wood 2005), etc.

· When treating logarithmic expressions or solving logarithmic equations, the

logarithm is eliminated or “cancelled out” by dividing by the symbol “

log

”

(Andelfinger 1985, p. 229; Berezovski 2004, p. 54; Kenney 2005, p. 4; Yen

1999, p. 6).

· In order to calculate the value of

logab

, the root

is extracted (Fermsjö

2014; Leopold and Edgar 2008; Weber 2002).

3.2.2 Difficulties in understanding the meaning of the logarithmic concept

In addition to the kinds of abortive algebraic manipulations listed above, some

authors also report on misconceptions in meanings. Even students capable of

correctly handling logarithmic expressions may labour under such

misconceptions:

· Students conceive logarithms as a “button […] on my calculator” (Watters and

Watters 2006, p. 280), as a special “number as Pi” or simply as a “maths

machine” (Weber 2013, p. 80f.).

· Students experience the problem of determining

log28

without a calculator as

more difficult than that of writing 8 as a power of 2 (Andelfinger 1985).

· Expressions like

loga1

and

logb1

(Kenney 2005) are thought to be different,

while, conversely, expressions like

log10 x

and

ln x

(Kenney 2005) or

log(x+3)

and

log x⋅log3

(Andelfinger 1985; Senk and Thompson 2006) are

not recognised as being different.

· Logarithms are not recognised as a suitable tool for mathematical decisions and

modelling. For instance, when asked which is the larger of the two numbers

25625

and

26620

, students may reason as follows: “

25625

is bigger because it

has a bigger exponent” (Berezovski 2004, p. 57). Conversely, some students

are unable to articulate the implications for an actual (chemical, biological,

physical) situation where two quantities are logarithmically connected (Watters

and Watters 2006; DePierro and Garafalo 2008).

Difficulties of this nature illustrate what failing to understand logarithms can

mean.

3.2.3 Various underlying causes

These difficulties are given different interpretations in the research literature. In

the literature surveyed for the purposes of this study, three different causes could

be identified:

· Students' prior knowledge (mainly the concept of powers or exponents) may be

insufficient, or the new concept may not be adequately integrated into it (Chua

and Wood 2005; Kenney 2005). More detailed analyses identify

misconceptions behind students’ algebraic mistakes such as

log(x+y)=log x+log y

and regard them as an overgeneralization of rules,

with a distinction being made between the overgeneralization of logarithmic

laws (Chua and Wood 2005), the overgeneralization of linearity (van Dooren et

al. 2008) or the overgeneralization of the distributive law of numbers to

operations with the symbol of “log” (Matz 1982). Other authors focus on the

students' visual perception or the visual characteristics of the algebraic

expressions. They explain overgeneralizations of the above type by the fact that

students “misperceive the problem situation” (Lee and Heyworth 1999, p. 228)

or by the “visual salience” of the algebraic transformations, i.e. a “visual

coherence that seems to make the left- and right-hand sides appear naturally

related to one another” (Kirshner and Awtry 2004, p. 242).

· The teachers may not make the background of the symbol sufficiently explicit

(e.g., Kenney 2005), or even evince an insufficient understanding of logarithms

themselves (Berezovski 2007). For example, Berezovski (2007) concludes in

her case study that many of the investigated pre-service secondary teachers had

insufficient subject matter knowledge and limited pedagogical content

knowledge regarding logarithms.

· The dominance of the indirect definition (e.g., Espedal 2015; Fermsjö 2014;

Mulqueeny 2012; Smith and Confrey 1994; Weber 2002; Williams 2011). In

one case study, Williams (2011) reports that her students used the Eulerian

definition as a kind of a one-way transferral to exit the new context of

logarithms, arguing and answering in the better known context of exponents

and not transferring the answer back.

Being rooted in the tradition of subject-matter didactics, this study concentrates on

this last cause and reconsiders the way logarithms are introduced and defined (cf.

3.4).

3.3 Alternative Instruction Proposals: an Overview of the Literature

Because of the many difficulties this topic can cause for many students, it is

sometimes discussed in teachers’ journals and explored in mathematics education

publications. These propose a number of alternative instructional approaches to

supplement or replace traditional instruction and which aim to avoid difficulties

and improve accessibility. It would appear that few of the alternative approaches

have undergone systematic quantitative evaluation or qualitative investigation, so

there is little empirical data regarding the learning of students who were taught

using these approaches. They will now be classified into five groups.

3.3.1 Using history

Most of the proposals go back to the historical roots of the concept, drawing on

Napier and Bürgi. They suggest introducing and explaining logarithms with two

juxtaposed progressions in the form of number lines or the columns of a table, and

teaching students how to reason on the basis of this model (Clark 2006; Fermsjö

2014; Katz 1997; Mulqueeny 2012; Smith and Confrey 1994; Toumasis 1993).

Only a few case studies have been conducted on the impact this kind of logarithm

instruction has on students (Fermsjö 2014; Mulqueeny 2012). Fermsjö (2014), for

instance, reports that this approach was helpful, and that difficulties in

manipulating logarithmic expressions were rarer. However, his students struggled

with some new systematic mistakes (ibid., p. 308).

Weber (not to be confused with the author of this paper) chooses a slightly

different approach, introducing the logarithm

logab

as the number of factors

. In a pilot study with two groups of university students, he explained the

logarithm to one group as the number of factors, while the other group received

traditional instruction (Weber 2002). Both groups were then asked to solve tasks

requiring basic computations (e.g., “What is

logxx

?”) and rules (e.g., “

loga(xr)

can be simplified to what? Why?”) (ibid., p. 1024). The treatment group

performed better on these tasks than the control group. Similarly, Espedal (2015)

developed some teaching material based on repeated division. This material was

presented to students of one high-school class, while another high-school class

received traditional instruction. The “repeated division” group solved tasks such

as “Why is

log1 =0

?” or “Solve

2x+1−3=5

” significantly better than the control

group (ibid., p. 56). However, several of the known mistakes in manipulating

logarithmic expressions still occurred in both groups.

3.3.2 Using alternative language and notation

As already mentioned, it is not immediately clear what is meant by “logarithm” or

“

logab

” (cf 2.1.). A second group of proposals therefore involves new language

and notation. Examples include the “index” (Brennan 2007, p. 49), the “exponent

seeker” (Bennhardt 2009, p. 48) or the “liftoff function” (Hurwitz 1999, p. 344).

A form of notation like

a!(b)

would borrow from propaedeutic algebra, where

variables are initially represented in the form of placeholders (Hammack and

Lyons 1995). One could also think about introducing the “tree notation,” a

notation adapted from linguistic theory and artificial intelligence (Kirshner and

Awtry 2004, p. 232). I did not find any literature that would propose the use of

historical notations such as or

⁝

(cf. 2.1.4).

3.3.3 Manual calculation

One pitfall of defining a logarithm as an exponent is that students cannot draw on

any pre-existing algebraic procedure to solve an equation such as

2x=10

Students mostly solve tasks like these with the “guess and check” strategy; the

teacher’s “solution”

x=log210

can feel like a cop-out and may easily become

entrenched as the “button on my calculator” idea (Watters and Watters 2006, p.

280).

To avoid problems like these, a third group of proposals suggests having students

calculate logarithmic values manually. This can be done using a sequence of

nested intervals, going back to Euler’s demonstration of how

log5

can be

calculated manually (Sandifer 2007). Other proposals suggest utilising slide rules

or logarithm tables (Berezovski 2004, p. 78; Ostler 2013).

3.3.4 Using different conceptualisations

Yet another group of proposals suggests utilising definitional properties other than

the Eulerian definition of logarithms (cf. 2.1.3). Some authors suggest starting

with Cauchy’s property, introducing logarithms as (continuous) functions that

fulfil the property

log(ab)=log(a)+log(b)

and deducing all the properties and

logarithmic laws step by step (Seebeck and Hummel 1959). Other authors use the

geometric fact that the area under the hyperbola is a logarithm (Panagiotou 2011),

going back to a proposal by Klein (1932, p. 156).

3.3.5 Applications

A final group comprises suggestions for applied, “real-life” examples. Teachers

present realistic situations, which logarithms can help to calculate (Klüpfel 1981)

or which are to be represented and explored with logarithmic scales (Kirsch 1977;

Rahn 1994; Wood 2005). More elaborated approaches like the Dutch Realistic

Mathematics Education use realistic problem contexts to introduce new concepts,

formalising only gradually. Thus to introduce logarithms, students are first

presented with the concept of exponential growth in several realistic contexts.

Only then are logarithms presented by interpreting the graph of a realistic

exponential function in reverse order (Webb et al. 2011). Following a teaching

experiment at a college, students claimed to have understood the meaning of

logarithms (ibid., p. 51). Whether they also had a better grasp of algebra is not

reported.

3.4 Conclusion

The many alternative teaching proposals on one hand (cf. 3.3), and the many

reported difficulties and mistakes on the other (3.2) suggest that it could be worth

reconsidering the traditional introduction of the concept (3.1). As mentioned

above, this paper takes the position that the indirect definition might be

responsible for many of the difficulties and mistakes, because learners cannot

immediately apply it. Here are some arguments:

1. Euler’s definition reflects a structural conception: Napier and Bürgi had a

certain understanding of the concept when they studied two juxtaposed

progressions. Although logarithms were recognised shortly thereafter to be

exponents, it took over a century until this interpretation ceased to be read as a

property and was used as a definition by Euler. In a way, it reflects the view of

an expert who, after a long period of research, retrospectively chose one of the

definitional properties as a definition, or, using Sfards terminoloy, it mirrors a

reification process that leads to a structural conception (1.2.1).

2. Later definitional properties are less operational: There are other ways of

defining and thus introducing the logarithm, for example as the solution of a

certain functional equation (2.1.3). However, this definition is purely

existential and thus even less operational than the Eulerian one. Although some

authors suggest this approach (Seebeck and Hummel 1959), it has been shown

to cause students great difficulty (Sajka 2003). With Kirsch, defining

logarithms as the solution of a functional equation is like “[…] putting the cart

before the horse, which serves only to make access more difficult […]” (1977,

p. 102).

3. Teaching logarithms is similar to teaching division: Although division can

logically be defined as inverse multiplication, it is — at least in the German-

speaking countries — not taught with such logical elegance. Instead, it is

conveyed through a wide range of contexts, namely as fair-sharing and

measurement and as repeated subtractions, leading to long division (2.2). The

ambition of these various approaches is to make division accessible, and to

convey an understanding of it as an autonomous object, possessing various

properties. Thus, since logarithms and division are mathematically analogous

(2.1.1), their teaching can be considered analogously as well: Introducing

logarithms in a more constructive way than as inverse exponents could

probably improve accessibility.

To my knowledge, this conclusion is not supported by any larger empirical study,

but is shared by other authors (3.3). Based on the theoretical construct of basic

models in the sense of ‘Grundvorstellungen’ (1.1), the next section will present

elements of a broader instructional approach that relies on multiple

interpretations, of which the interpretation as inverse exponent is only one.

4 Operational and structural basic models for

logarithms

If it is correct that the indirect definition is not a suitable starting point, this could

have implications for the teaching of logarithms: How, then, could the subject of

logarithms been structured differently?

Based on sections 1 and 2, it is possible to answer research question (Q1),

identifying four basic models for logarithms. All four will be shown to interpret

logarithms within the students’ world of experience in the sense of consolidated

school knowledge. To demonstrate their explanatory power, this section will show

how they can serve students as a basis for meaningful argumentation, i.e. how,

with their help, certain textbook tasks can be solved in a meaningful way and

certain typical mistakes and difficulties can be prevented (cf. 1.1.2). The

suggested courses of argumentation are of course prototypical ways of handling a

problem: They show one possible chain of argument. What is important here is

that learners are able to give similar reasons of their own that they find plausible.

Additionally, question (Q2) will be answered by elaborating the extent to which

each basic model can be used to explain logarithms operationally or structurally.

4.1 Logarithms as Multiplicative Measuring

Logarithms and divisions work analogously: While division asks for the number

of addends, logarithms look for the number of factors (cf. 2.1.2). In analogy to the

additive measurement model for division (1.1.4), multiplicative measuring is

proposed as a first basic model here. Reflecting Napier’s idea when he coined the

term “logarithm” and conceptualised it as number that counts divisions, a first

version of a basic model is as follows:

(BM1) The logarithm of a number

(to base

) indicates how often the base

fits into the number

as a factor. (Basic model of ‘multiplicative fitting-

into’)

Because counting the factors is equal to counting the divisions by the factor, it can

be slightly reformulated:

(BM1ʹ) The logarithm of a number

(to base

) indicates how often the number

has to be divided by the base

to produce

. (Basic model of ‘repeated

division’)

This version reflects the idea Burja and Schellbach have had in introducing their

notations modelled on division (2.1.4).

Mathematically, both interpretations are equivalent to the definition of logarithms

as inverse exponents because

can be read as the product of

factors of

. At

first glance, (BM1) and (BM1ʹ) work for natural numbers

only, but this is not a

principal restriction, because there is the logarithm algorithm: Firstly, it reflects

the basic model of multiplicative measuring as it divides by the base

repeatedly; secondly, it extends multiplicative measuring as it determines the

decimals of logarithms to any desired accuracy (2.2).

4.1.1 Multiplicative measuring as a basic model

To what extent can (BM1) and (BM1ʹ) be considered basic models in the sense of

‘Grundvorstellungen’ (cf. 1.1)? Both interpretations of logarithms refer to a

measuring context, but without indicating any realistic manual actions. That is

why they cannot be considered primary basic models. Now, because students have

compared and measured many realistic things since childhood in everyday and

geometric situations, they are well acquainted with linear measuring. Based on

this concrete experience, and combined with the possibility of calculating

(certain) logarithms manually, the question “How often does the factor

fit into

multiplicatively?” is quite likely to sound reasonable to students, at least more

so than the question about finding an inverse exponent. In this sense, (BM1) and

(BM1ʹ) are secondary basic models (cf. 1.1.3).

The explanatory power of (BM1) and (BM1ʹ) lies in their prototypical nature; they

can be used to tackle and solve textbook tasks. For example, consider the

evaluation of

log21024

. Students are usually expected to solve it by guessing and

checking. With the aid of multiplicative measuring, however, it can be

reformulated as “How many times does 2 as a factor fit into 1024?” (BM1) or as

“How often does 1024 have to be divided by 2 to produce 1?” (BM1ʹ). Thanks to

this interpretation, the answer can now be determined step by step:

log21024 =10

. This chain of argument can be depicted in a similar manner to

figure 1 (fig. 4).

What is the base 2

logarithm of 1024?

1024 ÷2÷…÷2=1024 ÷210 =1

∴!log21024 =10

(BM1, BM1ʹ)

Logarithms as

multiplicative measuring

How often does 2 as a factor fit into 1024?

How often does 1024 have to be divided by 2 to produce 1?

Fig. 4 Role of the basic model of ‘multiplicative measuring’ as intermediary between two contexts

of logarithms

Many further problems can be solved by relating them to the context of

multiplicative measurement, such as the simplification of

loga1

logaa

logaa3

etc., or checking the correctness of

logaab=b

: By reformulating the problem as a

question of how many times one number fits into another as a factor, it can be

plausibly solved. Assertions like “

log37

is less than

log57

because 3 is less than

5” (Berezovski 2007, p. 109) should also occur less frequently with this basic

model: As 3 is smaller than 5, 3 fits into 7 more often than 5, regardless of any

calculations. Even expressions like

log20

log2−8

can be recognised as

incalculable with the aid of this basic model, because

and

−8

can never be

converted to

through repeated division by

This basic model may perhaps even help to make the logarithmic law

loga(bc)=logab+logac

accessible: If

fits — as a factor —

times into

and

times into

, it has to fit exactly

m+n

times into the product

Empirical investigation is of course needed to establish whether students find this

argument plausible or whether it is too sophisticated.

4.1.2 Multiplicative measuring, an operational conception

The basic model of multiplicative measuring can be conveyed to students with the

aid of the logarithm algorithm (cf. 2.2). This consists of a set of computational

rules that can be performed step by step. As such, it constructively generates

results, and makes an operational access to logarithms possible.

4.2 Logarithms as Counting the Number of Digits

For certain kinds of questions and arguments, it can be helpful when basic models

relate a concept to a typical application context and vice versa (cf. 1.1.3). For

example, in the decimal system, one million has seven digits, and its common

logarithm is 6. More generally, the number of digits of any natural number

(in

decimal notation) is equal to

log10 n

⎢

⎣⎥

⎦+1

, so the concept of logarithm can be

interpreted as a number of digits. This interpretation can be generalised to non-

natural numbers of digits: Because a power

10n

has

n+1

digits, any number

can be said to have the generalised number of digits

log10 b+1

(in decimal

notation). In a nutshell:7

(BM2) The (common) logarithm of a number

finds the number of digits of

minus one. (Basic model of ‘counting the number of digits’)

4.2.1 Counting the number of digits as a basic model

Unlike the basic model of multiplicative measuring (BM1), this interpretation

draws on a typical application context for logarithms, the number of digits.

Students at upper secondary school are familiar with it from early maths lessons,

because counting and positional notation is addressed and practised at length in

primary school.

The potential of this basic model lies not in relating a class of logarithmic

problems to the context of the number of digits but, conversely, in answering

certain questions with the help of logarithms. For instance, consider a task like

“How many digits are there in the power

22000

when written in decimal

notation?” (Deller et al. 2000, p. 78; translation by C.W.). Using basic model

(BM2), it can be expressed differently, leading to a logarithmic expression. Unlike

the example discussed above, the corresponding cycle of argumentation now

begins in a familiar context (fig. 5). In this specific task, one furthermore has to

utilise the logarithmic law

loga(bn)=n⋅logab

(or (BM3), cf. 4.3), while the

decimals of

log10 2

can in principle be computed by hand (cf. 2.2).

How many digits are there in the

power

22000

when written out in

decimal notation?

The number therefore

has 603 places.

(BM2)

Logarithms as

counting the number of digits

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

finds the

number of digits minus 1 in the binary expression of

log10 22000 =2000 log10 2≈602.06

Fig. 5 Role of the basic model of ‘counting the number of digits’ as intermediary between two

contexts of logarithms

Other problems can also be solved with this basic model. For example, if tasked

with determining which of the numbers

25625

and

26620

is larger (Berezovski

2004), it is possible to compare the number of digits. Students familiar with this

basic model could be less likely to make statements such as “

25625

is bigger

because it has a bigger exponent” (ibid., p. 57). This is another claim that merits

empirical investigation.

4.2.2 Counting the number of digits, an operational conception

The basic models (BM1) and (BM2) are somewhat akin in that they both deal

with logarithms as numbers and place them in a familiar context. While the first

basic model refers to a constructive algorithm and thus emphasises the operational

side most strongly, the second basic model (BM2) does not. Instead, it deals with

an application of logarithms. So although this second basic model cannot be used

to explain logarithms operationally, it could be used to support operational

explanations in that it describes the effect it has on numbers. Because this differs

from mentioning the mathematical properties and relations of a concept, it cannot

be considered a form of structural explaining. In short, (BM2) can be considered

another operational basic model for logarithms as numbers, but to a lesser extent

than (BM1).

4.3 Logarithms as Decreasing the Hierarchy Level

While the law

logaab=b

is plausible on the grounds of the basic model of

multiplicative measuring (cf. 4.1), the logarithmic laws are not convincingly

evident on the basis of this alone. This requires a further basic model, not least to

avoid some of the algebraic difficulties involved in manipulating expressions and

solving exponential and logarithmic equations.

Arithmetic operations can be classified according to their hierarchy level:

Because, mathematically speaking, multiplication is — at least for natural

numbers — a repeated addition, it is a second-level operation (e.g., Oehl 1962).

Similarly, the operation of raising to a power is a repeated multiplication, which is

why it can be considered a third-level operation. Now, logarithms fulfil the

property

Φ(x⋅y)=Φ(x)+Φ(y)

(

x>0

y>0

) (cf. 2.1.3). As a consequence,

logarithms lower not only second-level operations (multiplication, division) by

one hierarchy level, but also third-level operations (powers, roots). To be more

precise:

(BM3) The logarithm of an expression reduces multiplications and divisions to

additions and subtractions, and it reduces powers and roots to

multiplications and divisions. (Basic model of ‘decreasing the hierarchy

level’)

4.3.1 Decreasing the hierarchy level as a basic model

The third basic model relates to the hierarchy of arithmetic operations as a

familiar context. Seen in this light, (BM3) refers to previous school experience at

primary and lower secondary level, where, for instance, multiplication is

explained as repeated addition. With the aid of this basic model, it is possible to

solve tasks like “expand

log cd

as much as possible” (Deller et al. 2000, p. 74;

translation by C.W.). As in the above scenarios, a possible chain of reasoning

answers the original task by reformulating it in another logarithmic context (fig.

6).

Expand

log cd

as much as possible.

∴!!!log cd =12(log c+log d)

(BM3)

Logarithms as

decreasing the hierarchy level

Taking square roots as a third-level operation becomes dividing the logarithm

by two, and multiplication as a second-level operation becomes addition

between logarithms.

Fig. 6 Role of the basic model of ‘decreasing the hierarchy level’ as intermediary between two

contexts of logarithms

This basic model can be used not only to solve similar problems, but maybe also

to make logarithmic laws like

loga(b÷c)=logab−logac

plausible and thus more

accessible. There is some theoretical plausibility to the claim that this basic model

could prevent students from making algebraic mistakes such as expanding

logarithmic expressions like

log(x+y)

log x+log y

(cf. 3.2.1). It might even

enable them to give reasons why this transformation must be wrong: For the

expansion of

log(x+y)

to be possible, there would have to exist a somehow

“deeper” arithmetic operation than addition, which is a first-level operation.

4.3.2 Decreasing the hierarchy level, an operational conception

In contrast to the former basic models, the third basic model is geared not towards

numbers, but towards algebraic expressions. As such, it interprets logarithms as

operators. Similar to the second basic model (BM2), it does not explain

logarithms operationally, but it likewise provides information about the effect of

logarithms. This is why it is theoretically plausible that this basic model could

support the operational explanation of logarithms in the case of algebraic

expressions, but not their structural explanation. In short, (BM3) can also be

considered an operational basic model, but for logarithms as operators rather than

for logarithms as numbers.

4.4 Logarithms as Inverse Exponents

Euler’s understanding of logarithms is widely used in mathematics, so it would be

intellectually dishonest and irresponsible to withhold it from learners. It therefore

has to be considered as one of several interpretations of logarithms, analogously

to inverse multiplication as one of the basic models of division (cf. 1.1.4). It is the

last basic model for logarithms proposed here:

(BM4) The logarithm of an expression (to base

) is the exponent by which the

base

must be raised to yield the expression. (Basic model of ‘inverse

exponent’)

4.4.1 Inverse exponent as a basic model

With its focus on inversion, this basic model again draws on previous school

knowledge. Students have encountered several inverse phenomena in the course

of their maths lessons: positive and negative numbers, multiplication and division,

etc. Each aspect was experienced as a phenomenon in its own right before it was

logically related to its inverse (cf. 1.1.4).

With the aid of this basic model, the law

alogab=b

is now self-explanatory: As the

logarithm of

is an exponent, the base

raised by this exponent must be equal

to the original number

. This basic model obviously also indicates how to solve

exponential equations: Because an equation such as

2x=40

asks for an exponent,

the question can be translated into another logarithmic context, so the task is to

determine

log240

(fig. 7).

Solve

2x=40

three significant figures.

Therefore,

25.32 ≈40

(BM4)

Logarithms as

inverse exponents

x=log240 ≈5.32

Fig. 7 Role of the basic model of ‘inverse exponents’ as intermediary between two contexts of

logarithms

4.4.2 Inverse exponent, a structural conception

This basic model refers to another concept, the concept of exponents, whether

numbers or expressions. Unlike the previous basic models, it neither implies a

computation procedure (as (BM1) and (BM1ʹ) do) nor names the effects

logarithms can have (as (BM2) and (BM3) do), but rather relates a concept to an

object. This is the view of experts who have studied the operational aspects of the

concept and then reified their experiences to make propositions about states and

objects. In other words, the Eulerian property reflects the most structural

conception of logarithms to date. As such, it is less suitable as a starting point for

learners than as the objective of concept formation at school (cf. 3.4)

5 Discussion and outlook

5.1 Summary: Four Basic Models, and Two Hypotheses Regarding

their Relevance

This work identifies various basic models to foster the learning of logarithms. The

main underlying theoretical construct is that of ‘Grundvorstellungen’ (Griesel

1971; vom Hofe 1998; vom Hofe et al. 2006; Oehl 1962). As such, it favours a

multi-perspective approach to mathematical concepts — i.e., that a mathematical

concept must be taught using a variety of different basic models —, an approach

that is known from the teaching of division (Brown 1981; Fosnot and Dolk 2001;

Greer 1992; Padberg and Benz 2011; Radatz et al. 1998). To theoretically

improve the accessibility of the logarithmic concept, the construct of

‘Grundvorstellungen’ is combined with another theoretical construct, namely that

of of operationality and structurality, which emphasises the importance of

operational approaches in the learning of new mathematical concepts (Sfard 1991,

2008).

The paper focuses on logarithms as numbers or operators. First, it begins by

unpacking some mathematical knowledge: Logarithms can be conceptualised as

inverse exponents (i.e. structurally). But they are more than just the opposite of

another concept;

logab

can also be seen as the number of factors

and thus

operationally. Because the indirect definition still prevails in algebra textbooks

despite the many problems reported in educational literature, it may be an

underlying cause of learners’ difficulties. This position is in keeping with other

authors (Fermsjö 2014; Mulqueeny 2012; Smith and Confrey 1994; Weber 2002;

Williams 2011).

The paper therefore develops several basic models in the sense of

‘Grundvorstellungen’: the basic model of ‘multiplicative measuring’ (BM1), the

basic model of ‘counting the number of digits’ (BM2), the basic model of

‘decreasing the hierarchy level’ (BM3), and the basic model of ‘inverse exponent’

(BM4). They all support meaningful mathematical reasoning, because they have a

certain explanatory power when it comes to argumentation or solutions to specific

tasks. All of them vary not only in terms of their mathematical character (number,

operator), but also in terms of their process-object character. They can therefore

be structured as follows:

· On one hand, (BM1), (BM1ʹ) and (BM2) refer to logarithms as numbers.

(BM3), however, refers to logarithms as operators, while (BM4) refers to both

numbers and operators.

· On the other hand, (BM1) and (BM1ʹ) are more closely related to the

operational conception of logarithms (as they generate result), whereas (BM4)

is more closely related to its structural conception (as it refers to another

concept). The remaining two basic models, (BM2) and (BM3), can potentially

be useful to bridge operational and structural conceptions of logarithms,

emphasizing the effects of logarithms.

The presentation of basic models for logarithms in this paper goes hand in hand

with hypotheses. In short, I claim that the interplay between the four basic models

can make logarithms accessible and foster understanding. This means: They

endow learners with a solid basis for argumentation, firstly to solve tasks and to

remember laws, and secondly to reduce mistakes and difficulties. Various

theoretical arguments are given in support of these two hypotheses.

5.2 Implications for Future Research

The primary purpose of this paper is to suggest elements of an approach that

makes logarithms accessible and meaningful for students. Many arguments are

made at a theoretical level here without a specific focus on empirical evidence.

However, preliminary observations based on my teaching experience suggest that

the four basic models may have the potential to help students generate meaning

about logarithms (Weber 2013). As encouraging as the corresponding indications

may appear, empirical research is needed to investigate the affordances and the

limitations of the basic models introduced here. For example: Does the plurality

of basic models for logarithms foster or hinder understanding? Is the sequence of

the basic models of importance for the teaching and learning of the concept? Is it

advisable, as Sfard and others suggest (cf. 1.2.1), to introduce logarithms using

the most operational conception (BM1), gradually moving to the most structural

one (BM4), or is it not advisable (cf. Gilmore and Inglis 2008)? A further concern

is that the multiple interpretations offered to students may not be reified (cf. Sfard

1991; 2008) as a new, single entity, but that learners may instead use them in a

compartmentalised fashion. Qualitative case studies, and quantitative large-scale

implementations, can help to explore and answer questions like these, and thus to

investigate the above hypotheses.

Secondly, the implications for teaching are not clarified for either school or

teacher education. For example, it is not obvious in what way and in which order

the various models have to be taught so that they can become reified in a single

object, the logarithm. Again, one could take inspiration in the teaching of division,

whether from textbooks for pupils (e.g., Wittmann and Müller 2004) or for pre-

service teachers (e.g., Padberg and Benz 2011; Radatz et al. 1998). Future

investigation must clarify to what extent such knowledge about instructing

primary-level pupils and teachers can be transferred to secondary-level students

and teachers, respectively. This is the second field where empirical research is

indispensable.

In summary, making logarithms accessible is a multilayered task. The work

reported here can be considered a first step towards this objective, laying some

theoretical foundations for further investigation.

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Dec 2024

In our research, we investigated how we can categorise the errors made by first-year university students in logarithmic tasks. We conducted a survey among first-year engineering students of computer science and logistics engineering at Neumann János University. A short self-made test was used for the research. Logarithmic errors were categorized into classes based on a two-dimensional categorization, the type of error and its possible source. We found that the two-dimensional categorization was able to classify most of the typical logarithmic errors listed in the literature, as well as new errors. As a result of our investigation, it became clear that it is not possible to categorise the types of occurring errors without identifying the underlying sources of these errors. As a general suggestion for similar studies, it is worthwhile to use multidimensional categorization, where one aspect tries to reveal the underlying source or problem solving strategy in addition to the visible error type.

Didaktisch orientierte Sachanalysen zur Ausdifferenzierung der allgemeinen Grundvorstellungen zum Funktionsbegriff – Exemplarisch durchgeführt an der SinusfunktionDidactically Oriented Subject Matter Analyses to Differentiate The General “Grundvorstellungen” of Functions—Exemplified by the Sine Function

Article

Full-text available

Jan 2025

Valentin Katter

Zusammenfassung Zu den allgemeinen Grundvorstellungen zum Funktionsbegriff zählen die Zuordnungsvorstellung , die Kovariationsvorstellung und die Objektvorstellung . Diese Vorstellungen können auf alle Funktionsklassen gleichermaßen angewendet werden. In jeder Funktionsklasse sind diese Vorstellungen allerdings mit charakteristischen Sachzusammenhängen sowie Denk- und Handlungsweisen verbunden, die wiederum als Ausgangspunkt neuer, differenzierter Grundvorstellungen dienen können. In dem vorliegenden Artikel wird eine Vorgehensweise vorgestellt, mit der eine Ausdifferenzierung der allgemeinen Grundvorstellungen zum Funktionsbegriff durchgeführt und funktionsklassenspezifische Grundvorstellungen hergeleitet werden können. Grundlage des Verfahrens bildet eine didaktisch orientierte Sachanalyse in Verbindung mit der didaktischen Phänomenologie Freudenthals und Aspekten der Genese mathematischer Begriffe. Angewendet wird dieses Verfahren schließlich auf die Funktionsklasse der trigonometrischen Funktionen, wodurch sechs normative Grundvorstellungen zum Sinus identifiziert werden.

Math-teur, a mathematical card game for (not only) fun

Article

Jan 2023

We present the game Math-teur, a mathematical card game used successfully at the John von Neumann University for teaching remedial mathematics as part of a complex university catch-up program. In the paper we describe the game itself and summarize the considerations used for the game design.

Factor Analysis of Students' Difficulties Understanding the Concept of Logarithms at SMA Negeri 2 Percontohan Karang Baru Aceh Tamiang

Article

Full-text available

Dec 2023

This research aims to determine the factors that cause students' difficulties in understanding the concept of logarithms using the factor analysis method. The samples taken were 120 class X students of SMA N 2 Percontohan Karang Baru Aceh Tamiang. Data was obtained using diagnostic test instruments for logarithmic material based on aspects of logarithms as objects, as processes, as functions, and in contextual problems. From these four aspects, Author 2 was able to identify 20 variables that caused students difficulty in understanding the concept of logarithms. The research results show that there are three factors that cause students' difficulties in understanding the concept of logarithms. These factors are (a) logarithmic expressions (b) logarithmic functions, and (c) logarithmic calculations. The logarithmic expression factor is the dominant factor which has an eigenvalue of 10.282 which is able to explain a variance value of 54.115%.

Senior High School Mathematics Teachers’ Perception of the Teaching and Learning of Logarithm in the Kwahu East District

Article

Full-text available

Dec 2023

The motive of the study was to find factors that influence the teaching and learning of logarithms at the senior High schools in the Kwahu East Municipality. The study employed a descriptive survey where a questionnaire was administered to Eighty-Six (86) teachers. The sample was selected using the Census method. After the SPSS software was used to analyse the data, the descriptive analysis revealed that most teachers positively perceive logarithms. Factors influencing Senior High School mathematics teachers' teaching of logarithm were identified. Teachers believe that suitable teaching methods can influence students' perceptions. Strategies to encourage positive perceptions should be promoted among teachers, and recommendations include developing innovative methods for teaching logarithms in Senior High School mathematics.

Historische, logische und individuelle Genese der Trigonometrie aus didaktischer Sicht

Book

Full-text available

Jan 2023

Valentin Katter

Action-Process-Object-Schema Analysis of Students' Conceptual Understanding of Logarithms

Article

Full-text available

Oct 2024

This study is a qualitative multiple-case study research aimed to analyze the students' conceptual understanding of logarithms using APOS (Action-Process-Object-Schema) theory. Using the framework, we analyzed interview results of eight math students with varying performance levels. It revealed that despite a common progression of understanding of logarithms, two important APOS mechanisms show the point of divergence among performance groups. One is the efficient transformation of logarithms to exponents (interiorization), and another is the relation of the logarithm process with other important processes such as functions and equations (coordination). Both mechanisms are important to see logarithm as an object, more than an operation needed to be solved by the student. This result provides insights for teachers in developing APOS-based lessons that can be implemented in the classroom.

Mathematical Grundvorstellungen in Chemistry Education: An Approach to Understanding Chemical Mathematics Using the Example of the Concept of Density

Article

Oct 2024

Sharing Teaching Ideas: A Simple Way to Teach Logarithms

Article

May 1995

We have taught logarithms in high school and college algebra and in introductory calculus courses. Before we began using the method of teaching logs described in this article, we found that many students had difficulties mastering the concept, more so than with other functions. Other teachers reported similar experiences.

Using Logarithms to Explore Power and Exponential Functions

Article

Mar 1994

Power functions and exponential functions often describe the relationship between variables in physical phenomena. Power functions are equations of the form y = kx ⁿ (see fig. 1 ), where k is a nonzero real number and n is a nonzero real number not equal to 1. Exponential functions are equations of the form y = kb x (see fig. 2 ), where k is a nonzero real number and b is a positive real number. Students should be able visually to recognize these functions so that they can easily identify their appearance when experimental data are graphed. When physical phenomena appear to describe exponential and power functions, logarithms can be used to locate approximate functions that represent the phenomena.

When Are Logarithms Used?

Article

Apr 1981

Charles Kluepfel

With the widespread use of pocket calculators, no one would use a table of logarithms to multiply or raise a number to a power. Of course, the calculator raises numbers to arbitrary powers by using logarithms internally, but why should students learn logarithms if they don't plan to design calculators? Certainly the presence of the log key on many calculators will do away entirely with the need for tables of logarithms. But when is the log key to be used?

Sharing Teaching Ideas: We Have Liftoff! Introducing the Logarithmic Function

Article

Apr 1999

Marsha Hurwitz

The Curriculum and Evaluation of Standards for School Mathematics (NCTM 1989) targeted the concept of function as one that should receive increased attention. A function can be defined as a rule that assigns to every input value exactly one output value. To make it easier for novices to recognize a function, we should select the rule, the input, and the output to be straightforward.

Logarithmic and exponential functions —a direct approach

Article

Oct 1959

A direct approach to logarithms rather than the indirect approach as an inverse of an exponential function has many advantages for teaching secondary school students.

Instrumental and relational understanding

Article

Jan 1976

R. Skemp

“Grundvorstellungen” as a Category of Subject-Matter Didactics

Article

Jun 2016

One of the central issues that has long captivated research efforts in mathematical education concerns the question of what mental representations people have of mathematical content. That is, what meaning do people associate with mathematical content, and what role do these mental representations play in teaching and learning mathematics? To this end, a number of different categories concerning mental representations have been developed, including, e. g. “intuition”, “use meaning” or “concept image”. One approach, stemming from the German Stoffdidaktik (subject-matter didactics) tradition, is constituted by the didactic category of Grundvorstellung(en). In this contribution we begin by investigating the origins and the development of the concept of Grundvorstellung in the German-speaking research literature on mathematics education. Then, we summarize the main ideas of the concept, particularly concerning its normative, descriptive, and constructive use as well as the distinction between primary and secondary Grundvorstellungen. Following this, a number of typical areas of application will be considered. We will then discuss the use of Grundvorstellungen in the construction and classification of items for quantitative assessment, as well as the role of Grundvorstellungen as a descriptive and explanatory category in empirical classroom research.

DIE ENTWICKLUNG FUNKTIONALEN DENKENS IN DER SEKUNDARSTUFE I – VERGLEICHENDE ANALYSEN UND EMPIRISCHE STUDIEN ZUM MATHEMATIKUNTERRICHT IN DEUTSCHLAND UND FRANKREICH

Thesis

Jan 2008

Stölting,, Pascal

Les Dépendances fonctionnelles sont omniprésentes dans la vie quotidiennes, mais les résultats de plusieurs études montrent que les élèves ont des difficultés à traiter des problèmes de ce domaine. Ce travail compare l'approche des dépendances fonctionnelles en France et en Allemagne, à l'exemple de la Bavière.Dans la première partie la pensée fonctionnelle est définie clairement et est reliée aux cadres théoriques pertinents utilisés en France et en Allemagne, comme les Grundvorstellungen (Vom Hofe), les registres sémiotiques(Duval) ou le concept image (Vinner). Les outils nécessaires pour les analyses de ce travail y sont également développésLes chapitres suivants sont dédiés à la comparaison des programmes et des livres de classe. Le but est d'éclaircir dans quelle mesure les élèves des deux pays sont aidés dans le développement de la pensée fonctionnelle.Puis il est essayé d'identifier en pratique les points forts et les points faibles reconnus lors des analyses des curriculums. Deux approches différentes ont été choisies pour étudier comment les élèves travaillent avec la pensée fonctionnelle et quels problèmes surgissent lors de ce travail: - des analyses quantitatives des données de PISA et PALMA sont faites et amènent à démontrer certaines relations avec les résultats des analyses précédentes.-Les analyses quantitatives d'une étude d'interviews réalisé en France et en Allemagne documentant certains points forts et points faibles identifiés auparavant. Finalement des propositions sont faites sur la bases des résultats de ce travail pour essayer d'éviter les points faibles des deux pays et pour chercher à profiter des points forts.

Aspects of simplification in mathematics teaching

Chapter

Jan 1977

A Kirsch

On the Generation of Basic Ideas and Individual Images: Normative, Descriptive and Constructive Aspects

Chapter

Jan 1998

Rudolf vom Hofe

According to present-day knowledge, the process of mathematical problem solving, even at the university level, is always combined with intuitive images and assumptions which unconsciously affect the outcome. This has been shown, in particular, by Fischbein, in several investigations (1979, 1983, 1987, 1989, 1990). There seems to be no thinking without images, in this sense. At best these images can affect mathematical thinking positively. But they can also lead us astray — as Fischbein has shown as well — if inadequate images become fixed as unconsciously effective ‘tacit models’. Therefore the question arises how to deal with this intuitive level. For instance, does one assume that adequate images develop automatically with an appropriate formal contact with mathematics, or does one consciously follow and support the generation of adequate images and ideas?

Making Logarithms Accessible — Operational and Structural Basic Models for Logarithms

Abstract and Figures

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