In: T. Dooley & G. Gueudet (Eds.) (2017), Proceedings of the Tenth Congress of the European Society for Research in

Mathematics Education CERME10 (pp. 537–544). Dublin, Ireland: DCU Institute of Education and ERME.

Multiple models for teaching logarithms: with a focus on graphing functions

Christof Weber

School of Education, University of Applied Sciences Northwestern Switzerland, Liestal, Switzerland; christof.weber@fhnw.ch

This argumentative paper focuses on graphing logarithmic functions and presents some mathemati-cal knowledge for teaching, employing vom Hofe’s construct of basic models (or ‘Grundvorstel-lungen’) and Sfard’s distinction between operational and structural conceptions. On the assump-tion that difficulties students have with graphing might be a consequence of the standard interpreta-tion of logarithms as inverse exponents, it claims that logarithmic functions should not be intro-duced structurally, as inverse exponential functions. Instead, several operational models of the log-arithmic concept are proposed, and their explanatory power for graphing logarithmic functions is expounded. These models are intended to serve students as a meaningful basis for argumentation.

Keywords: Logarithmic functions, basic models, operational—structural, knowledge for teaching.

Introduction Logarithms as functions differ from logarithms as numbers (e.g. determining log21024 ) or opera-tors (e.g. using logarithms to manipulate expressions and solve equations), especially when it comes to teaching and learning (Smith & Confrey, 1994). To make logarithms as numbers and operators meaningful and thus more accessible to learners, some subject matter knowledge for teaching has been developed by Weber (2016). This was achieved by combining the theoretical construct of basic models (or ‘Grundvorstellungen’, vom Hofe & Blum, 2016) with the construct of operational-ity and structurality (Sfard, 1991; 2008). The present paper focuses on the functional side of loga-rithms and its teaching, namely on the sketching of logarithmic graphs. After recalling various con-ceptualisations of logarithms and analysing some students’ difficulties in graphing logarithmic functions, four basic models are discussed with regard to their explanatory power for graphing loga-rithmic functions. Because of the mathematical analogy between logarithms and division (cf. We-ber, 2016), some arguments used here are analogous to those familiar from the teaching of division (Ball, Thames, & Phelps, 2008; Greer, 1992; vom Hofe & Blum, 2016).

Background Various conceptualisations of the logarithmic concept

As we are interested in different ways of viewing logarithms, it is worth taking a quick look at the history of mathematics. Several properties of logarithms have been discovered since their invention in the early 17th century (for details, see Weber, 2016):

(P1) Napier and Bürgi conceptualised logarithms as numbers that count divisions. For example, log2 8 equals 3 because 8 has to be divided by 2 three times (to yield 1). In analogy to the standard division algorithm, this interpretation can be extended to a logarithm algorithm for the manual calculation of logarithms proceeding by repeated division down to 1 instead of re-

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peated subtraction down to 0: one divides instead of subtracting, exponentiates by 10 instead of dividing by 10, trying to yield 1 instead of 0 (ibid., pp. 79–80).

(P2) The conception found in current collections of formulas only became possible after Descartes had invented the symbol for powers, an. This notation enabled logarithms to be seen as parti-cular exponents, a property that Euler used in his 1765 definition loga b = x :⇔ ax = b . More-over, the left-hand equation can be read not only arithmetically (focusing on the number x ), but also functionally, taking “ log ” as a function of the argument b .

(P3) Another conceptualisation was provided by Cauchy in the 19th century. He proved that loga-rithmic functions are, apart from a factor, the only continuous solutions Φ of the functional equation Φ(x ⋅ y) =Φ(x)+Φ(y) ( x > 0 , y > 0 ). Specifying the logarithmic function in implicit form, this conceptualisation is a purely functional one.

Each of the properties (P1) to (P3) highlights a particular aspect of logarithms. Because, from a ma-thematical perspective, they are all equivalent, each property could serve as a definition of the con-cept. From an epistemological perspective, however, their qualities are distinct, as discussed below.

Logarithmic functions, their graphing and some of the difficulties students face

Logarithmic functions are essential in calculus and for modelling processes, which is why they are taught at secondary level and in undergraduate courses. As with other types of functions, there are several challenges for students, such as when it comes to interpreting graphs verbally, or deciding whether or not a given graph represents a certain function (e.g. Leinhardt, Zaslavsky, & Stein, 1990; Markovits, Eylon, & Bruckheimer, 1986; 1989). For instance, one well-known student misconcep-tions says that any function should be linear (e.g. Sfard, 2008, p. 21). An issue specific to logarith-mic functions is the confusion of the logarithmic graph with the “combined graph” (Kastberg, 2002, p. 129), i.e. with the exponential and the logarithmic graph merged into a single image. A similar misconception reported is students viewing the graphs of y = 2x and y = log2 x as being “exactly the same” (Williams, 2011, p. 54). Misconceptions like these might be caused by the standard in-troduction of logarithmic functions as inverse exponential functions.

As space precludes other aspects, this paper deals with the graphing of logarithmic functions only, i.e. with manually sketching the graph of a function in a Cartesian plane, based on its logarithmic equation y = f (x) . To analyse some of the difficulties that can arise here, the following steps (S1) to (S4) involved in graphing logarithmic functions are identified:

(S1) Determining and calculating an appropriate number of pairs (table of values): This step in-volves selecting a finite series of x -values that lie within the domain of the function, which in the case of logarithmic functions is a proper subset of the real numbers. In particular, one has to determine a first x -value, a last x -value, and the pattern the series follows (e.g., if there is a pattern, whether the difference or the ratio of two consecutive x -values is constant).

(S2) Drawing the axes and scales, and plotting the corresponding points (“local construction,” Leinhardt et al., 1990, p. 13): The plotted points form the supporting points of the graph to be drawn.

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(S3) Connecting the supporting points with a line segment (“prediction,” ibid., p. 13): This step involves making a conjecture based on the visual characteristics of the plotted points. It will therefore be referred to here as “graphical interpolation”. It includes decisions regarding the curvature and the degree of smoothness of the line (e.g. differentiable at the points).

(S4) Extending the graph to the right and left of the line segment: This is a second prediction to make, based on a part of the graph, referred to here as “graphical extrapolation”: Does the graph straighten out to a straight line? If not, in what way does its curvature change? Is it bounded, does it have vertical asymptotes, intercepts with the axes, etc.?

In each of these steps, the lack of a meaningful basis for argumentation can result in difficulties: Firstly, determining the domain in (S1) is a known issue (e.g. Markovits et al., 1986, 1989). As an illustration, Figure 1 shows two students’ graphs of the function y = log2(x)−3 . Student A’s graph extrapolated to the left intersects the x -axis and thus exceeds the domain. Moreover, his x -values form an arithmetic progression, which is not optimal in terms of the corresponding y -values.1

Figure 1: Students’ documents showing their tables of values for f (x) = 2x+3 and g(x) = log2(x)− 3 ,

together with the corresponding graphs (left side: student A, male; right side: student B, female)

Secondly, graphs are sometimes thought of as isolated points (Leinhardt et al., 1990), or the sup-porting points may be interpolated with a straight line (Markovits et al., 1989). Figure 1 shows that in their third step (S3), both students chose the graphical interpolations to be a more or less straight line, at least piecewise. And thirdly, in (S4), the graph is sometimes not extended beyond the range of the supporting points at all, or only by a little; or there may be an extrapolation to one side which suggests a progressive growth of the logarithmic function (cf. student B in Figure 1).

1 The two first- and second-year undergraduate students (18 and 19 years old) had attended a precalculus course held at a public university on the east coast of the USA in spring 2016. Logarithmic functions had been introduced in the tradi-tional way, i.e. as inverse exponential functions. The documents shown in Figure 1 are from their final examination. The exact wording of the task was: “Consider the two functions f (x) = 2x+3 and g(x) = log2(x)−3 : a) For each func-

tion, create a table of values, choosing your x -values carefully. b) Graph both functions on the same set of axes.”

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Students’ difficulties like these give rise to the following question: What mathematical knowledge for teaching logarithmic functions could endow learners with a meaningful basis for argumentation in order to potentially reduce their difficulties?

Understanding functions and logarithms

There are many ways to conceptualise what it means to understand the concept of a function in gen-eral (e.g. Lauritzen, 2012; Markovits et al., 1986; Sfard, 2008), or the concept of logarithmic func-tions in particular (Berezovski & Zazkis, 2006; Kastberg, 2002). Interestingly enough, graphs are rarely included in conceptualisations of how functions are understood; when they are, they are used to gauge whether it is possible to derive the equation of a certain function (Markovits et al., 1986).

Graphing as a vital aspect of understanding functions

Focusing on the opposite — graphing equations — here, Lauritzen’s (2012) conceptualisation of procedural and conceptual knowledge of functions is useful, because he attaches importance to the construction of graphs. To measure the ability to perform “graphic procedures” (ibid., pp. 52–53), he asks students to sketch the graph of a function, thus subsuming graphical interpolation (S3, see above) and extrapolation (S4). On the other hand, he considers calculating values (S1) a type of “al-gebraic procedure” (ibid., pp. 54–55). In Lauritzen’s theoretical framework, graphic and algebraic procedures together operationalise the procedural knowledge of functions. In other words, graphing can be seen as a vital part of understanding functions.

Operational and structural conceptions of functions, and the discourse on functions

Sfard’s analysis of how mathematical notions are formed shows that conceiving mathematical no-tions (1991) and talking about them (2008) can happen in two fundamentally different ways: as processes (operationally), or as objects (structurally). For instance, learners tend to read equations of functions and tables of values operationally, whether as prescriptions of how to calculate values of the function, or as a covariation between two quantities (Sfard, 1991, p. 15). By contrast, they tend to perceive graphs of functions structurally, as “[…] infinitely many components of the func-tion […] combined into a smooth line, […] as an integrated whole […]” (ibid., p. 6). According to Sfard (1991), firstly, operational approaches are more accessible to learners when forming new con-cepts. Secondly, concept formation, for instance of functions, means subsuming the discourses on equations and graphs in a new discourse. For example, the concept “logarithmic function” is reified as soon as the discourse on logarithms as numbers and operators is merged with the discourse on logarithmic graphs, as soon as they “become mere representations” (Sfard, 2008, p. 122).

As such, the history of the logarithmic concept reminds us of the reification of rational numbers (Sfard, 1991, 2008): On one hand, because the conceptualisation (P1) of logarithms as numbers that count divisions can be transferred to a set of computational rules that make it possible to calculate logarithmic values step by step, it expresses an operational view (similar to the division 1÷ 2 ). This is why Sfard’s findings on functions could apply to logarithmic functions as well, as her analysis of understanding functions relates to polynomials only, and their operational character. On the other hand, a historically more recent conceptualisation such as inverse exponents (P2) expresses a struc-tural view (similar to the fraction 1 2 ).

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Basic models for logarithms as numbers and operators

To help students access and understand a certain mathematical concept, it is sometimes embedded in a context that is realistic or, if this is not possible, in a context that is at least familiar to the stu-dents (cf. the Dutch “realistic mathematics education”, van den Heuvel-Panhuizen, 2003). In the German-speaking countries, basic models is a theoretical construct to capture what is meant by making concepts accessible, or understanding them (referred to as “Grundvorstellungen”, vom Hofe & Blum, 2016). Put simply, a basic model for a concept must have two characteristics: Firstly, it is an interpretation of that concept in a context in which students are likely to have more experience, and secondly, it has a certain explanatory power, that is, it is flexible enough to be applicable to dif-ferent mathematical situations. For instance, when division is seen within the everyday context of fair-sharing, an equation such as 30 ÷1 2 = 60 is difficult to follow or perform. However, within the context of splitting-up or measuring, it can be explained as “1 2 fits into 30 sixty times”. Both basic models of division, fair-sharing and splitting-up, are thus indispensable for understanding di-vision (referred to as “partitive” and “quotative division”, Greer, 1992). For Ball and colleagues, they constitute the specialized content knowledge for teaching division (Ball et al., 2008, p. 400).

For the teaching of logarithmic functions, no basic models are known thus far. For logarithms as numbers and operators, however, I have previously identified four models (for details, see Weber, 2016):

(BM1) Logarithms as multiplicative measuring: The logarithm of a number b (to base a ) indicates how often the base a fits into the number b as a factor. This interpretation derives from the algorithm mentioned above (P1), or from the relation b / alogab =1 . Example: log21024 can be simplified to 10 because 2 as a factor fits into 1024 ten times. As it generates result, mul-tiplicative measuring emphasises the operational side of logarithms most strongly.

(BM2) Logarithms as counting the number of digits: The (common) logarithm of a number b finds the number of digits of b needed to represent b in positional notation, minus one. This in-terpretation derives from the fact that the number of digits of any natural number n (in dec-imal notation) is equal to log10n⎢⎣ ⎥⎦+1. Example: The number 22000 has 603 digits when

written out in decimal notation because log1022000 ≈ 602.06 . In describing the effect it has

on numbers and thus dealing with a specific application, this interpretation could be used to support the operational explanation of logarithms in the case of numbers.

(BM3) Logarithms as decreasing the hierarchy level: The logarithm of an expression reduces third-level operations (powers, roots) to second-level operations (multiplications, divisions), and it reduces second-level operations to first-level operations (additions, subtractions). This inter-pretation derives from property (P3). Example: The expression log cd can be expanded to 12 logc + logd( ) because the taking of square roots, as a third-level operation, becomes di-

viding by two, and multiplication of the variables becomes addition of their logarithms. In describing the effect it has on expressions and thus dealing with another specific application, this interpretation could support the operational explanation of logarithms for expressions.

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(BM4) Logarithms as inverse exponents: The logarithm of a number (or expression) to base a is the exponent by which the base a must be raised to yield the number (or expression). This derives from property (P2), and is useful for solving exponential equations. Example: 40 as a power of 2 is approximately 25.32 because log240 ≈ 5.32 . Because this interpretation re-lates logarithms to another object (exponents), it reflects the structural view of experts.

In the next section, this collection of operational and structural basic models will be shown to have the potential to serve as a basis for argumentation for the graphing of logarithmic functions as well.

Basic models for logarithmic functions and their explanatory power From a mathematical standpoint, every property of logarithmic graphs can be derived from expo-nential graphs, using (BM4) as a basis for argumentation. From an epistemological viewpoint, how-ever, conceiving logarithmic functions as inverse exponential functions reflects the structural view of experts who have reified their experiences, and not the view of learners. Perhaps, as Sfard (1991) and others suggest, operational conceptions should instead precede structural ones as consecutive steps to be passed through when teaching a new concept such as logarithmic functions? As basic models (BM1) to (BM3) do not replace logarithmic functions with another class of functions, they could be more appropriate for learners than model (BM4). Instead, they inform students about what logarithms “do” and what logarithms “are good for”, interpreting them within contexts in which learners are likely to have some experience (counting, long division, having an effect on …, etc.).

In what respect, then, could the three basic models described offer students a meaningful basis for argumentation when they are introduced to graphing logarithmic functions? And in what way could they potentially reduce the difficulties described above? Here are some arguments:

1. Operational conceptualisation: The logarithm algorithm, which stems from property (P1), “loga-rithms are numbers that count divisions”, and is captured by basic model (BM1), can allow stu-dents to conceive a logarithmic equation such as f (x) = log2(x)−3 operationally, much like a polynomial one: “First, calculate how often the base 2 fits into x by repeated division, then sub-tract 3.”

2. Domain of the function: In order to graph a logarithmic function, it is essential to determine its domain (cf. (S1) and Figure 1). With reference to the model “logarithms as multiplicative meas-uring” (BM1), values such as log(0) and log2(−8) can be recognized as incalculable because there are no reasonable answers to the corresponding questions “How many times does 10 as a factor fit into 0?” and “How many times does 2 as a factor fit into –8?”; neither 0 nor –8 can be converted to 1 through repeated division. This is why logarithms of 0 and of negative values can-not be defined.

3. Pattern of the finite series of x -values: Another choice to be made for graphing a logarithmic function easily is the pattern that the series of x -values follows (S1). In view of the basic model “logarithms as decreasing the hierarchy level” (BM3), the x -values should follow a geometric series, with the ratio of two consecutive x -values equalling the base. The logarithm would then transform the geometric series into an arithmetic one, resulting in equidistant y -values.

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4. Growth of the graph: As we have seen above, both graphical interpolation and graphical extrapo-lation can cause many problems (cf. (S3), (S4)). Referring to the basic model “logarithms as counting the number of digits” (BM2), the growth of logarithms can be recognized as non-proportional: In general, doubling a number does not double its number of digits. Furthermore, it is strictly increasing and unbounded above because this is how the number of digits behaves. Thus neither the interpolated nor the extrapolated graph can be a straight line, but must increase monotonically, growing degressively.

Discussion This work builds on my earlier paper about the basic models for logarithms as numbers and opera-tors (Weber, 2016). There, the supposition was discussed that the standard textbook explanation loga b = x :⇔ ax = b could be too compact or “dense” for many learners to serve as a meaningful basis of argumentation, which may be why dealing with logarithms often turns into mere manipula-tion of formal symbols, causing students’ difficulties (ibid., pp. 85–86). If this applies to logarith-mic functions as well, an alternative, broader way of introducing and teaching logarithms is re-quired. For this reason, this paper discusses some content knowledge for the teaching of logarithmic functions. The guiding theoretical construct is that of basic models (vom Hofe & Blum, 2016), combined with the construct of operationality and structurality (Sfard, 1991, 2008). The four basic models, developed previously for logarithms as numbers and operators (Weber, 2016), are shown here to have some explanatory power for logarithmic functions and their graphing, that is, that they could potentially help to make logarithms meaningful and reduce some common difficulties that students encounter.

This paper lays some theoretical foundations for future research. To what extent an approach with multiple basic models can facilitate more meaningful teaching and understanding in the actual classroom will have to be investigated carefully. There has been a first encouraging episode from my own teaching, where a student who in general struggles with mathematics realized why loga-rithmic functions cannot be proportional: Making use of basic model (BM2), not the standard inter-pretation (BM4), she argued precisely as in point 4 in the previous section. A teaching experiment is therefore planned in the near future to study the affordances and limitations of the basic models, exploring the discourse of students who are taught not just one but four interpretations of the loga-rithmic concept. In analogy to the teaching and learning of division with multiple models, a crucial point will be the students’ shift from the multiple basic models proposed here to the object of loga-rithmic functions. Or, to cite Freudenthal (1975, as quoted in van den Heuvel-Panhuizen, 2003, p. 15, italics in original): “Models of something are after-images of a piece of given reality; models for something are pre-images for a piece of to-be-created reality”.

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