’Literature’ on mathematical modelling from a teacher perspective: A textbook’s portrayal

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’Literature’ on mathematical modelling from a teacher perspective: A textbook’s portrayal

Anders Wolfsberg

To cite this version:

Anders Wolfsberg. ’Literature’ on mathematical modelling from a teacher perspective: A textbook’s portrayal. CERME 9 - Ninth Congress of the European Society for Research in Mathematics Educa- tion, Charles University in Prague, Faculty of Education; ERME, Feb 2015, Prague, Czech Republic.

pp.951-957. �hal-01287273�

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a teacher perspective: A textbook’s portrayal

Anders Wolfsberg

University of Copenhagen, Faculty of Science, Copenhagen, Denmark, rnp123@alumni.ku.dk

In this paper, it is proposed to reformulate the question of whether modelling as depicted in academic litera- ture is insufficiently implemented in school. Rather, the question proposed is: what form of modelling is actually portrayed in textbooks and curricula – understood as teachers’ ‘literature’ on mathematical modelling?

For illustration, a case is offered wherein perspectives on modelling are traced from academic literature, through curriculum, and an upper secondary textbook, all taken from a Danish context. It is discussed how the textbook’s portrayal of mathematical modelling deviates from that in academic literature. Further, it is suggested that this deviation – this gap – may be closed by fitting the version of modelling to be implemented better to the institution- al structure of upper secondary school.

Keywords: Mathematical modelling, textbook portrayal, connecting research to practice.

INTRODUCTION AND DISPOSITION

Large scale implementation of mathematical modelling (hereinafter modelling) activities in mathematics education is a visionary project that has been under way for the past several decades. According to many researchers it is still present in mathematics classrooms to a degree far from satisfactory (Barquero, Bosch, & Gascón, 2010).

The aim of this paper is to understand better the causes and consequences of the apparent gap between research on modelling, and modelling as taught in practice.

One may take the position that researchers have a particular ‘vision’ of a subject to implement in practice – their version of modelling. Observing that it is not satisfactorily implemented, one is led to the question:

‘what are the obstacles hindering its implementation?’

This paper proposes a change of perspective.

Instead, the viewpoint taken is that a subject known as mathematical modelling exists in the upper secondary school institution, one that is different from the subject conceived of in academic research. Hence, it is integral to find out what this subject is, how it relates to academic perspectives, and why it takes the form it does. This might guide the way in conceiving of a version of modelling that is better fit to be implemented in teaching-in-practice in school classrooms.

A case is presented wherein a particular notion of mathematical modelling is traced from academic literature, throughout curriculum documents to a textbook’s portrayal. All three samples constituting the case are taken from a Danish context.

The case is taken as point of departure for a discussion of the differences between two versions of modelling:

that which is portrayed in academic literature, and that portrayed in textbooks and curricula – ‘literature’

implemented in secondary school.

DIDACTIC TRANSPOSITION

Theory of Didactic Transposition (TDT) is a theoret- ical framework developed by Yves Chevallard. TDT describes the transition of scholastic knowledge produced in universities throughout the educational system, e.g., curriculum and textbooks, to teaching situations. The mathematics being taught is not the same as that being produced in research institutions.

Indeed, academic knowledge naturally undergoes changes when transposed through curriculum into textbooks (Winsløw, 2011).

The external transposition process takes place in what is called the ‘noosphere’ where ‘scholarly knowledge’

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‘Literature’ on mathematical modelling from a teacher perspective: A textbook’s portrayal (Anders Wolfsberg)

952 is transformed into ‘knowledge to be taught’, to be

found in, e.g., curriculum and textbooks.

The first step [the external transposition] corresponds to the study of the formation of the ‘teaching text’ (…) and highlights the conditions and constraints under which the ‘knowledge to be taught’ is constituted (…) (Bosch & Gascón, 2006, p. 56)

The internal transposition describes how teachers adapt and implement the knowledge into the very teaching situation (Winsløw, 2011).

Didactic knowledge such as that on teaching math- ematical modelling and promoting modelling compe- tency is produced in communities of mathematics and educational science researchers. However, the knowledge emerges in curriculum and textbooks (often by the work of others) in a transposed form more or less dissimilar to that envisioned by educational researchers.

Thus, in a TDT view, incoherencies between knowledge of (some) researchers, and that portrayed in textbooks, and applied by teachers, is a naturally occur- ring phenomenon. Indeed, constructing a so-called epistemological reference model (hereinafter reference model) of knowledge on a given topic in all levels of the didactic transposition allows for analysing these incoherencies (Winsløw, 2011).

METHOD OF ANALYSIS

The case study consists of three samples of ‘knowledge on mathematical modelling’: one from academic literature, one from curriculum, and one from a textbook. The contents of these samples are analysed qualitatively. In view of TDT this method corresponds to constructing a three-level reference model for

‘knowledge on modelling’ – a view of modelling in three different ‘spheres’ of the educational system.

Hence, only the external transposition is analysed in this case.

All three samples are taken from a Danish context but are of international relevance. Thus, the scope on literature focuses on Danish researchers who work with mathematical modelling and the concept of modelling competency as used and developed by Mogens Niss.

Niss has had significant influence on the wording of

Danish mathematics curriculum via the KOM project on mathematical competencies (Blomhøj & Kjeldsen, 2010). Furthermore, interpretations of the modelling subject such as that of Niss are widely used in international literature on modelling (see, e.g., Niss, Blum,

& Galbraith, 2007). The textbook is one of the most widely used in Danish upper secondary mathematics teaching.

In the following three sections, views on mathematical modelling as portrayed in academic literature, curriculum, and textbook are extracted.

Inspiration has been drawn from Julie and Mudaly (2007), who identify two categories for tendencies in didactical research on teaching modelling: ‘modelling as vehicle’ and ‘modelling as content’. In a modelling as vehicle perspective, modelling is conceived of as providing motivation and support for learning of other mathematical subjects. As content, modelling is conceived of as a topic in itself.

In the following analysis and extraction of academic views, and curriculum and textbook descriptions, the focus is set on the category of modelling as content.

Thus, content is sampled that describes modelling as educational content, not as a way of teaching. What is excluded are views on how modelling could or should be taught.

ACADEMIC VIEW ON MODELLING Models and modelling

Niss (1989) defines a mathematical model as a triplet (A,f,M) of three domains denoting an extra-mathematical domain, A, a mathematical domain, M, and then a mapping, f, between the two. The domains must be understood abstractly as ”collections of relationships, phenomena, questions (and possible answers) and such-like” (Niss, 1989, p. 28).

Niss identifies modelling with model construction, and in defining the concept, he characterises the process that signifies general model construction processes. We find an equivalent characterisation of the modelling process in Blomhøj & Kjeldsen (2010, pp. 3–5), illustrated in Figure 1 – the modelling cycle.

The process is illustrated cyclically since the object to be modelled often is redefined progressively. Between the sub-processes are double-headed arrows to indi-

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cate that activity in any one sub-process influences the entire model in either direction simultaneously.

The sources of information for the model are shown in the three ellipses.

Niss (1989) points out, on the nature of models, that

”Models are designed to model something – not to be confused with something unique.” (p. 31).

Modelling competency

The modelling competency is one of eight mathematical competencies identified by Niss in the KOM Project which are characteristic to mathematical activity. Mathematical competency is described as

“someone’s insightful readiness to act in response to the challenges of a given situation” (Blomhøj & Jensen, 2007, p. 47). Specifically, the modelling competency is characterised as:

A person’s insightful readiness to autonomously carrying through all aspects of a mathematical modelling process in a certain context and to reflect on the modelling process and the use of the model (Blomhøj & Kjeldsen, 2010, p. 3)

Thus, “insightful” and “reflect on” indicates that, for teaching purposes, abilities beyond mere mechani- cal skills are sought in students. The student must be able to construct models, and “autonomously” at that.

Furthermore, it is emphasised that the student in this regard be able to consider a model and the process of constructing it holistically, not simply as separate algorithmic steps. Finally, the student must be capable of reflecting on “the use of the model”, which can be understood as having capabilities for both internal and external reflections:

Internal reflections add meaning and quality to the sub-processes involved in a mathematical modelling process, (…) the external reflections address the role and function of the model in actu- al or potential applications. (Blomhøj & Kjeldsen, 2011, p. 1)

Purposes

In (Blomhøj, 2009, p. 6) the main purposes for teaching and learning mathematical modelling, as given by the above definitions, are divided into three categories.

Below these are described in a shortened, partly re- formulated, but content preserving form:

― Proficiency: To learn modelling, it being a subject of the mathematical discipline and to provide motivation and support for learning of other mathematical topics.

― Applications: To apply mathematics in dealing with the challenges of private and professional life.

― Cultivation: To strengthen students’ competences as critical and insightful participants of a highly technologically developed democratic society.

MODELLING AS PORTRAYED IN CURRICULUM In this section references to modelling and models in Danish curriculum for mathematics Stx A, B and C levels are explained (Danish Ministry of Education [DME], 2013, Bilag 35–37). The levels A, B, and C refer to upper secondary mathematics education. A level is the highest.

Figure 1: Left: the modelling cycle/process. Right: the six sub-processes (Blomhøj & Kjeldsen, 2010, p. 4)

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954 Under “disciplinary goals” (DME, 2013, Bilag 35, 2.1)

models and modelling are mentioned several times.

Exact quotes are not given here, but are presented in synthesised form in Table 1 illustrating the results of the analysis.

The curriculum subsection “Supplementary material”

proposes teaching modelling on B level, and differen- tial equation models on A level. This supplementary material must “provide perspective and depth to the central material” (DME, 2013, Bilag 35, 2.3).

Under “Interaction with other disciplines” it is point- ed out that material must be included so that the student can understand “the significance of considering and discussing the preconditions for any mathematical description of reality, and the validity of results obtained hereby” (DME, 2013, Bilag 35, 3.4).

MODELLING IN THE TEXTBOOK

The B level mathematics textbook ”Gyldendals Gymnasiematematik Grundbog 2B” (Clausen, Schomacker, & Tolnø, 2007b) contains the chapter

“Models” (Danish: “Modeller”) of 37 pages, wherein modelling and models are treated.

The entire textbook series, this book included, intro- duces models when treating a diversity of other subjects (variable correlations, functions, etc.). Many of the examples given there are revisited in the analysed section, focusing now on their specific qualities as models and the construction of these by modelling.

In the following, Clausen and colleagues (2007b) is referenced whenever only page numbers are stated in parenthesis.

The textbook chapter commences with a motivation- al text giving a number of examples of models (nine pages). Thereafter follows the theory on modelling (two pages). It is emphasised that “The mathematical model describes a real-world situation” (p. 91) that

“may have limited range” (p. 91), and that it can “give insight into, and overview of, the real-world situation that is described” (p. 91).

A simplified version of the modelling cycle is shown illustrating the sub-processes “Translation to mathematics”, “mathematical treatment” and “translation to reality”. These steps are presented in a non-cyclic,

algorithmic manner, connected by single-headed arrows (p. 91).

Throughout the succeeding two pages, examples and exercises – by references to the exercise book (Clausen, Schomacker, & Tolnø, 2007a, pp. 27–33) – are given on model analysis. In particular, emphasis is put on identifying variables and parameters, respectively, and on their range: “Consequently, (...), x must take values between 200 and 1200” (p. 92).

The succeeding 21 pages contain more examples and exercises, now with emphasis on constructing models – modelling. All six sub-processes from Figure 1 are visited in linear succession in nearly every example. With several examples it is clarified that a model can be generalised: “Considering an alternative thick- ness for the paper sheet (...)” (p. 92), and “It is easy to generalise the model to considering alternative widths of the canal (...)” (p. 98). However, no examples or exercises are given wherein a point is made of progressively redefining real-world situations to be modelled. The approach is algorithmic rather than holistic.

The exercises focus on the student activity of extract- ing and interpreting information from a model, as well as constructing models. As with the examples, no exercises are given wherein the student must assess or reflect on the quality or validity of a model.

The remaining three pages of the chapter consist of a text considering existing “professional” models, and how their construction and use depend on applicants’

interests. For example: “An oil company and an environmental organisation will hardly construct and interpret a model in the same way.” (p. 116).

WHAT IS MODELLING? THREE POINTS OF REFERENCE

In this section the content extracted in the above three sections is organised and illustrated in Table 1. The table is organised so that distinct aspects of modelling may be viewed comparatively in terms of their depiction in each of the three levels of the educational system analysed.

Firstly, extracts from the analysis have been sorted under either purposes of applications and proficien- cy, or, under the purpose cultivation (see “Academic

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view on modelling” above). Secondly, four colours;

purple, red, green, and blue highlight the four main comparable aspects of modelling that I wish to em- phasise. These four points focus on portrayals of the modelling cycle (purple), the modelling process (blue), internal critical reflections (red), and external critical reflections (green).

DISCUSSION

This discussion sets out first to explore what versions of modelling are portrayed in the specific case elabo- rated above. This is done relative to the scope on academic literature on modelling that the case presents.

Subsequently, possible causes for the gap between modelling in academic literature and textbooks are discussed. Lastly, it is suggested that a version of modelling better fit for implementation might be conceived of by considering the institutional structure of secondary school.

Lack of holistic and critical perspectives

In the textbook only examples and exercises are given where models are considered as results of following certain mathematical calculations in linear order (see blue and purple coloured text in Table 1). This approach is predominantly algorithmic as it never invites students (and teacher) to consider the model holistically: perspective is in every example and exercise confined to sub-processes of the modelling process.

Note that a holistic understanding of the modelling process was emphasised in literature (as presented in the case) for being essential in developing auton- omous competencies. This is so since autonomy is understood as deriving from seeing both what is to be done to solve a problem, and why. An algorithmic viewpoint hence confines students and teacher to considering what step to do next, not what it means for the model in its entirety. As a consequence, the form of modelling portrayed in the textbook does not promote well autonomous competences.

Also, critical competences are underemphasised (see red and green coloured text in Table 1). In the textbook, assessment of a model’s validity and range amounts merely to examining ranges of variables and parameters. This is a devaluing interpretation of the curriculum statement on understanding the “range of models”. Indeed, consider the notion of critical reflec- tions in the scope of the case on academic literature. In this view, “range of models” is understood as what the model in its entirety explains, its descriptive power and its limitations, not simply the range of variables used in it.

A text example is offered in the textbook describing how an oil company and an environmental organisation might interpret a model differently. Assessing others’ use of models, which academic literature em- phasises as part of modelling competency, is therefore somewhat an element of the modelling subject por-

Literature Curriculum Textbook

Purpose declarations

-Applications - Proficiency

- Knowledge of math’s usability in formulating and treating problems from other disciplines

No indication

Specific points of note

- Complex modelling cycle with double-headed arrows

- Holistic perspective on modelling process

- Internal critical reflections on models and model construction

- Skills in constructing and us- ing models (geometrical, statis- tical, simulations)

- Knowledge of basic properties of models and modelling - Reflection on idealisations and range of models

- Linear modelling cycle with single-headed arrows

- Iterative modelling process. No examples or exercises display- ing a holistic perspective - Examples and exercises on range of variables in models Purpose dec-

larations

- Cultivation - Knowledge of mathematics’ interaction with culture, science, technology

No indication

Specific points of note

- External critical reflections - Reflection on assumptions and validity of mathematical descriptions (under “interaction with other disciplines”)

- Text example of an oil company and an environmental organisa- tions’ different interpretations of a model

Table 1: A case of ‘mathematical modelling’ in three levels of the educational system

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956 trayed in the textbook. Yet, no techniques to making

these critical model assessments are offered.

Hereby, model criticism seems but a matter of examining ranges of functions and variables, not assessing its significance for the real-world situation it is sup- posed to describe. Thus, the cultivation purposes of academic literature, and the corresponding purpose declarations of the curriculum are underemphasised, if present at all, in the textbook. That is, the use and role of models in decision making is hardly part of the modelling subject portrayed in the textbook.

A fragmented version of modelling

The lack of holistic and critical perspectives in the textbooks indicate an overall conception of modelling as a mathematical technique to solve mathematically stated problems, not real-world problems. This may seem natural as it is a topic presented in a mathematics textbook. Yet, it is worthwhile to ponder the consequences of portraying this mathematics-centred version of modelling.

In the scope on academic literature presented in the case, holistic and critical perspectives are integral aspects of the modelling process and competency. These aspects, however, necessitate an approach to modelling that goes beyond mathematical algorithms and concepts. When transposing the subject into teaching material for mathematics class proper, we might sus- pect that the subject that results becomes centred on the mathematics. That is, aspects that go beyond that discipline become underrepresented. We saw in the above that this suspicion seems justified in the case studied.

Under which subject, then, do the holistic and critical aspects of modelling sort, if not under the subject known as modelling in mathematics textbooks?

The curriculum phrasing indicates a hint. Note that it is stated in Table 1 (in green) that reflections on assumptions and validity of mathematical descriptions sort under ‘interaction with other disciplines’. Thus, it seems reasonable to assume that these aspects of modelling are considered content to be taught in interdisciplinary work.

Hence, it is reasonable to suggest that modelling in upper secondary are in fact two subjects: a mainly mathematical one found in mathematics textbooks,

and one pertaining to extra-mathematical aspects found in interdisciplinary teaching material. This might seem reasonable as the teacher can use different material – different literature – when teaching modelling.

Yet, assuming that modelling should be a subject that in its very nature aims to connect the mathematical and the extra-mathematical, this fragmentation is problematic. Indeed, reflecting on the extra-mathematical situation that one aims to model, constructing a suitable model, and then assessing its validity is a process of connecting mathematical and the extra-mathematical domains.

However, this situation is only problematic when seen relative to academic perspectives on modelling.

Modelling as a subject seems to find its own form in practice, one that fits the institutional structure, e.g., with subjects divided in disciplines and activities considered to be either mathematical or interdisciplinary. Understanding better this version of modelling-in-practice, and the processes that determine the form it takes is what this paper has aimed at.

CONCLUSION AND PERSPECTIVES

This study set out to investigate the gap between research on modelling and its degree of implementation in school. Rather than viewing the gap as an issue, it was considered a condition of connecting research to practice.

From this viewpoint, the question was not which obstacles lay in the way of implementing modelling in practice, and how they may be overcome. Rather, it was to understand how, and why, the subject of modelling takes the form it does in textbooks and curriculum.

Departing from a specific case, it was discussed how modelling as portrayed in textbooks and curricula differs from modelling as envisioned by researchers.

In particular, it was indicated that this subject, modelling-in-practice, is fragmented into two subjects. One to be taught in mathematics class proper focusing on the mathematical work in modelling activities, and one aimed at interdisciplinary work focusing on critical reflections and holistic aspects of the modelling process.

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It was argued that this fragmentation is a natural consequence of the division of activities common in secondary school, between mathematical and interdisciplinary activities. Taking this situation to be a condition rather than an issue (a viewpoint that must also be taken!), it is reasonable to ask whether the version of modelling to be implemented must be reshaped into a form more fit to the common secondary school structure. Indeed, could research provide a version of modelling that in distinct ways targets mathematics class proper, interdisciplinary work, as well as activities that bridge the two?

To be sure, researchers must attempt to implement the version of modelling that they see fit, and ensure that the teaching institution is changed accordingly to accommodate this subject. Yet, the point made in this paper is that investigating the version of modelling that actually occurs in teaching materials, one might see that the particular vision of modelling one has could be fitted better to the institutional structure of school.

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