Modeling as a process in Mathematics Education

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CHAPTER III MODELING AND CONTEXTUALIZATION IN SKETCHPAD

Introduction

In this chapter a philosophical background to modeling is presented to underscore the view that our views about the nature of mathematics influence our methodological and tool choices in the classroom. Various philosophical positions are discussed in a nut shell and their consequences for mathematics instruction evaluated. The meaning of a ‘model’ is revisited and further elaborations made influenced largely by the Realistic Mathematics Education philosophy of the Freudenthal Institute in the Netherlands. Various perspectives on modeling are examined as explicated in the mathematics education literature in an attempt to situate modeling within the mathematical, cognitive and didactic contexts. Modeling is then situated in a dynamic mathematics learning environment where Sketchpad is used as a modeling and simulation tool to enhance the understanding of the derivative concept. The modeling perspectives are threaded into the conjectured teaching/learning trajectory presented in the previous chapter in line with the chosen philosophical direction (compare 2.5). The various perspectives on modeling are then synthesized and an integrated meaning of modeling mooted.
Ultimately modeling is cast as a teaching strategy to permeate and characterize the envisaged didactic practice entirely. In a sense this chapter aims at answering the following research question: What instructional model and didactical relationships are conducive to a successful orchestration of dynamic mathematics software to enrich students’ concept image of the derivative?

Philosophical background to modeling

3.2.1 The Importance of a Philosophical View in Mathematics Education Philosophical views influence perspectives about methodological and tool choices. Dossey, McCrone, Giordano and Weir (2002:8) remark that the conception of mathematics held by teachers significantly influences how they teach it. This is clearly an echo of Rene Thom’s assertion:
Whether one wishes it or not, all mathematical pedagogy, even if scarcely coherent, rests on a philosophy of mathematics (Thom, 1973:204).
This suggests, among others, that the teacher’s understanding of the manner in which mathematical knowledge is acquired inevitably influences the way (s)he teaches it. Brousseau’s (1997) analysis of mathematics teaching supports Thom’s assertion. Brousseau’s notion of the didactical contract explains how the role of mathematics teacher is shaped by its institutional context. In his view, the role of the mathematics teacher is defined and shaped by the responsibility of teaching mathematics and the justification that any activity (modeling) in mathematics classrooms must include an explanation of how the activity is mathematical. Freudenthal’s (1991:14-15) view of mathematics as an activity buttresses the individual learner’s contribution, ‘his/her activity’ in the mathematical learning process, which includes not just reading, listening, reproducing mathematics as given, but also the aspects of producing mathematics and coming up with his/her own products. These products are largely representations, which are in themselves models achieved through the activity of modeling. In both senses, the pedagogy of the mathematics classroom rests to a lesser or greater degree on a philosophy of mathematics.

Logicism and Mathematics Education

Strauss (2001:19) notes that although it may seem natural to relate mathematics as a special science to the aspects of number and space in the first instance, the logicist, Russell wants to stress that mathematics is not concerned with quantity, but with order. A teacher with a logicist view might, as Russell aspired, want to reduce all of mathematics to logic, by declaring that mathematics and logic are identical. The view of mathematics as critically concerned with order is shared by Hamilton who defined algebra as the science of pure time and order (Cassier in Strauss, 2001:19). Cassier held the view that the main purpose of the study of the history of mathematics is ‘to illustrate and confirm the special thesis that ordinal number is logically prior to cardinal number, and more generally that mathematics itself can be defined, in Leibnizian fashion, as the science of order (ibid).
The whole goal of mathematics learning and modeling could then be to establish order. But then we are quickly reminded of retorts to these claims, as evidenced by the formalists and, more recently, the failure of the New Math programme, which attempted to rest all school mathematics on set theoretic logic with disastrous consequences. Logicism had to concede that it failed in providing a successful reduction to logic of the notion of infinity because the logic of infinity was not considered to be an axiom of logic. Yet a clear understanding of the completed/actual infinity underlies the notion of derivative as a limit.

Formalism and its influence on Mathematics Education

A teacher with an axiomatic formalist view of mathematics might, as Hilbert did, argue against the logicists that no science can exclusively be based or couched in logic hence:
Mathematics has a guaranteed content independently of all logic … there is a further prerequisite for the application of logical conclusions ….namely that something must be given in the conception: specific extra-logical objects intuitively present as immediate experience prior to all thinking (Strauss 2001:39).
It is clear that although formalism arose as a response to logicism, it is strong in its desire to axiomatize and formalize. Formalism works on the foundations of set theory, and those dealing with the philosophy of mathematics, often refer to mathematics as “the science of formal systems”. This suggests that to the axiomatic formalist teacher mathematics is set theory. These inclinations are precisely a didactic problem in the sense that they lead to an emphasis of starting with the end-products of mathematics: definitions, theorems, etc which becomes an antididactical inversion as observed earlier. Robinson (1967:39) points out that Cantor, as founder of set theory, was convinced that set theory deals with the actual infinite. This suggests that a teacher defining mathematics as set theory places the problematic dichotomy between the uncompleted/potential infinity and the completed/actual infinity at the heart of the definition. His/her practice will be coloured and refracted through the same prism. This formal view does not seem to be in tandem with a modeling perspective to mathematical instruction because pure mathematics with no consideration of any applications, invariably excludes applied mathematics and mathematical modeling.

Platonism and its Influence on Mathematics Education

Regarding the relationship between the universal and the individual, platonic realism designates ‘objective reality’ to an independent/universal existence outside the knowing human soul/individual. A teacher with a platonic view might, therefore, present mathematics as a structure ‘existing outside the mind and experiences of the student’ (Wessels 2006:5). If mathematics is out there, ready made and waiting to be discovered, then it still leaves room for the practitioner to guide students to rediscover mathematics for themselves during the learning process. The platonic view would therefore not necessarily favour the transmission model of teaching which takes the learner to be an empty vessel to be passively and unresponsively filled up. That would not necessarily fly in the face of instrumenting computational tools into artifacts with which to model mathematical phenomena.

Positivism and its influence on Mathematics Education

Positivism’s epistemological and philosophical idolization of the experimental method on the basis of sensory perception blocks out insight (thought) or practical reason as a possible source of mathematical knowledge by means that transcend the domain of sense perception and logical understanding. Strauss (2001:53) notes that by the end of the nineteenth century and the beginning of the twentieth century positivism emerged as a philosophical trend with the explicit purpose to abolish whatever supersedes sense perception. While a teacher who subscribes to positivism might therefore adopt a rigid formal naturalistic approach to methodology, such a possibility cannot just be uncritically attached to ‘positivism’ as it may equally apply to an experimental or empirical approach. In a sense it is more about a view of learning math that influences teachers’ decisions. Again such a teacher would may not necessarily be inclined towards the transmission model as there are many things that impact on teachers’ decisions and choices of a certain approach, only one of which is their philosophical view of math. A teacher who holds an empirical/empiricist view may teach directly in the transmission mode, not because of their philosophical view but because they believe from their understanding of how students learn but that they learn quickest. Another teacher with a formalist perspective, may choose to slowly guide and develop students through different stages, to finally cut the ontological bonds, and to study math as an axiomatic system, divorced from the contexts from which they were originally developed.

Aristotelian views and their influence on Mathematics Education

Aristotle believed that our knowledge of the individuality of entities closely coheres with the way in which we experience the identity of those things (Strauss, 2001:74). That is, this identity is something given to us in our experience and can therefore not be construed or accessed via other modes of knowing reality. A teacher with an Aristotelian view of mathematics might, therefore, rely heavily on experiences, and perhaps to the detriment of developing relations in a more abstract and general setting (Wessels 2006:7). Yet some of the greatest scientific breakthroughs have been achieved by defying commonsense intuition of the experienced world. For example, the Copernican discovery that it is the earth that goes round the sun and not the sun around the earth as daily commonsense experience suggests is a classic case in point. Galileo’s ex-communication by the church symbolizes how intolerant an Aristotelian teacher might tend to be towards students experimenting with new mathematics or unconventional ways of representing mathematical concepts in an environment laden with new digital technologies.

Empiricism and its Influence on Mathematics Education

Empiricism, is a (neo-) positivist philosophy which, with its stress on experience as a source of knowledge, designed a scientific methodology which begins with particular sensory data/sensory impressions on the one hand and the logical construction of entities from these impressions, on the other. This resulted in the well-known progression of empirical perception-hypothesizing-testing (verification) as the accredited methodological approach to theory formation (verified hypothesis) (Strauss, 2001:3) An empiricist teacher, thus, who prefers total reliance on experience-based learning might, like the Aristotelian, be more inclined to deprive students the opportunity to abstract and generalize to the formal level of mathematical cognition, yet it should be a question of when and how to get them there. Murray, Olivier and Human (1993:73) criticize the empiricist view of teaching as the transmission of knowledge and learning as absorption of knowledge and advocate that students should be given opportunities to construct their own mathematical knowledge. However, they do so from a learning theory point of view, rather than a philosophical point of view.

Constructivism and its Influence on Mathematics Education

Within the foundations of mathematics ‘constructivism’ means something completely different from ‘constructivism’ as a theory of knowledge and knowledge acquisition. In relation to the first, constructivism rejects the law of the excluded middle (and therefore all proofs based on contradiction (*). Its goal is to systematize mathematics without having to prove the existence of objects without showing how they can be constructed, hence the name “constructivism”. The second constructivism, which is of interest in this study, comes from general philosophy and can be described as an epistemology of how knowledge is gained (**). In the latter sense, Ernst von Glasersfeld’s basic principles of radical constructivism are the following:
1. Knowledge is not passively received either through the senses or by way of communication, but it is actively built up by the cognizing subject.
2. The function of cognition is adaptive and serves the subject’s organization of the experiential world, not the discovery of an objective ontological reality. (von Glasersfeld, 1988:83)
Von Glasersfeld (1993) acknowledges that his principles are built on the ideas of Jean Piaget, who applied the biological concept of adaptation to epistemology. He refers to his ideas as “postepistemological” because his radical constructivism posits a different relationship between knowledge and the external world than does traditional epistemology (Johnson, 2008:1). By emphasizing the knower as an active cognizing agent the radical constructivist teacher might have links to teaching and learning in ways that are an alternative to the transmission model and therefore supportive of modeling. Conversely, by denying the objective, mind-independent existence of the world, the radical constructivism might entail a return to a magical, capricious world-view that might stifle knowledge growth Hom (2000:156) and therefore might appear to be against mathematical modeling.
As a theory of learning social constructivism recognizes that mathematical knowledge is a product firstly of an individual human activity and secondly a social activity in that the individual’s subjective knowledge must be shared with others to become accepted objective knowledge or joint activity (Hurme & Jarvela, 2005). Furthermore students are expected to construct or model their own knowledge according to prior understandings, which envelop and colour their interpretation of new knowledge. This philosophy is therefore promotive of experimentation in the classroom. Lagrange (2005:147) points out that although experimentation is a basic choice in physics teaching it is more and more mentioned in Mathematics teaching as well. This suggests that in class the teacher has to present the observational basis (‘the experiment’) to the students without theorization and make theorization appear as built from experiment. This enables the classroom to be both a scientific research institution devoted to knowledge production and a didactic institution devoted to apprenticeship. In other words the constructivist views the learner as an active participant in the construction of his own models and processes of understanding.

An analysis of the meaning of a model

Lesh and Doerr (2000:362) observe that in physics, mathematics, chemistry or other physical sciences, a model is a system consisting of elements, relationships among elements, operations that describe how the elements interact and patterns or rules (e.g. symmetry, commutativity, transitivity, etc). A model in this sense has a structure made up of components and relationships and dynamism of operations that interrelate or connect the workings of the component elements one to another to represent a physical reality. We can view this as an explanatory role of a model. Within a problem solving context, Lesh, Hoover, Hole, Kelly and Post (2000:598) consider a model to be a simplified description that focuses on significant relationships, patterns, and trends. It simplifies the information in a useful form, while avoiding or taking into account difficulties related to surface-level details or gaps in the data. This seems to be a descriptive view of students’ ways of thinking.
Freudenthal (1991:34) refers to a model as an intermediary by which a complex reality or theory is idealized or simplified in order to become accessible to more formal treatment. It enables one to grasp the essentials of a static or dynamic situation by discovering common features, similarities, analogies, and isomorphisms towards the goal of generalizing. In this sense a model is seen as a simplified (generalized) representation of the structure and dynamics of a complex situation. He rejects the term ‘mathematical model’ in a context where it wrongly suggests that mathematics directly or almost directly applies to the environment. Rather a model can be considered to be an instrument or tool by which to gain clearer understanding of an otherwise complicated situation. English and Halford (1995:13) note that in cognitive science a model is a hypothesized knowledge structure and processes underlying the learning and application of mathematics. This view of a model shifts attention towards a different kind of object to be modeled – from physical phenomena or situations to modeling of thought processes learning scenarios and applications of mathematical knowledge to solve real world problems. This conjures up a problem-centredness dimension to modeling, a characteristic that is in harmony with the Realistic Mathematics Education vision.
Mudaly (2004:85) shares a similar view when he regards a model as a theory of the way the learner thinks, processes, checks solutions, makes a plan and executes it. This view mixes up the psychological processes of mental models or schemes with Polyan problem solving as aspects of modeling. By contrast, Wessels (2006) more categorically considers problem solving to be a simplified version of modeling which suggests that it is a sub-set of the larger modeling process and cannot, therefore, be equivalent to it (compare with Aristotle’s whole-part relationships in 2.2.2).
Taken together these definitions and views on modeling broaden the scope of modeling in this study to include how mathematical knowledge (the concept of derivative) can be effectively represented (the teaching dimension) and its meaning negotiated (student and teacher roles) in Sketchpad (the modeling tool), to deepen students’ understanding (concept images) of the derivative so that they can competently and confidently solve problems which require this knowledge (applications). In this study therefore modeling extends beyond the cognitive activity of formulating mathematical models, to a more general teaching methodology, and to how learners can make mathematical meaning in a dynamic mathematics environment and how these meanings and understandings can be represented, adjusted and refined.

Modeling as a process in Mathematics Education

Mathematical modeling and the Scientific Method

According to Dossey et al (2002:114) a mathematical model is a mathematical construct designed to study a particular real-world system or phenomenon and includes graphical and symbolic representations, simulations and experimental constructs of a model. This characterization is compatible with Sketchpad’s capabilities but limited in that verbal and numerical representations are not explicitly included but left implied. The real-world dimension is echoed by Chaachova and Saglam (2006:16) who view modeling as indicating the translation of a real phenomenon to a model, analysis of the model and a translation back to reality. This view is in natural synchrony with the context of discovery and invention of the derivative (rate of change of motion system) referred to in Chapter II (see 2.2.3).
The mathematical modeling process of Dossey et al (2002) is itself further characterized as a cognitive system (see Figure 3.1). Data gathered from the real world are converted to a model, which in turn is analyzed and mathematical conclusions drawn. Sketchpad, for instance enables the user to convert data into data plots of the relationship between two variables, x and y, which can be linked to produce a graphical representation to facilitate analysis and drawing of conclusions (compare 2.8.4). In the context of this study such conclusions are the determination of the rate of change of a functional relationship.
According to Dossey et al (2002) these conclusions of the modeling process can then be used to make predictions and explanations, which are tested against real-world data (phenomena) again to complete the modeling cycle. Given a value of x, the plotted function can be used, to predict a corresponding y-value to inform us how any y-(instantaneous rate of change) will vary with respect to x. Additional samples of real world (or simulation) data can be drawn and plotted again (compare 2.8.4) – to check the graph model against real world data to complete the modeling cycle.
Dossey et al (2002) further summarize the model construction process into six steps (see Figure 3.2). The first step is problem identification (in real world data or phenomena), which should be sufficiently precise so that it is translatable into mathematical statements. In the second step assumptions about hypothesized functional relationships between variables are made. For instance, hypothesizing whether the relationship is linear, cubic, quadratic, or curvilinear. The third step interprets the mathematical model by putting together all sub-models to see what the model is telling us as the ‘best’ or ‘optimal’ solution. The fourth step verifies or tests the model before use to check if it is answering the problem faithfully and sensibly. That is, a check is made as to whether the model makes sense, is reasonable or can be corroborated. The fifth step implements the model and applies it in a user-friendly manner. That is, the model is trial run. In the sixth and final step the model is maintained as changes occur. It is clear that the six steps can be supported by Sketchpad, in so far as the functional relationship being dealt with in this study is concerned i.e. continuous functions, differentiable functions, etc.
The six-step model construction process has clear reminiscences with the Scientific Method in that both make assumptions or hypotheses, gather real-world data, and test or verify hypotheses using that data (Dossey et al, 2002:118). The similarities or parallels confirm modeling as a scientific and objective undertaking. However, Dossey et al (ibid), are of the opinion that modeling and the scientific method differ in their primary goals. Whereas the goal in modeling is to hypothesize a model through evidence (real world data) the objective is neither to confirm nor deny but to test the model’s reasonableness or plausibility. Many functions to be differentiated in introductory calculus are not necessarily an exact mathematical representation but an idealization or model. Hence there is compatibility between modeling and the mathematical objectives of using Sketchpad.
A relevant feature of Sketchpad is that even in the absence of real-world data students can graph/plot functions, which they can use as examples to acquaint themselves with the structure and shapes of the wide range of functional relationships that are possible between physical phenomena. In other words, Sketchpad not only facilitates modeling by way of fitting real world data (in fact it is less suitable for this) but more importantly it offers model fitting opportunities to real world data or situations by offering a repertoire of graphical, numeric and dynamic representations for hypothesized symbolic relationships between variables.

A systems perspective of the modeling process

In their elaboration of a model as a system, Lesh and Doerr (2003:362) contend that for a system to be a model it must be used ‘to describe some other system or to think about or to make sense of it or to explain it or to make predictions about it.’ (Lesh & Doerr 2003: 362). There seems to be some equivalence with the scientific model perspective of Dossey et al discussed above, in terms of sense making reasonableness of representation, and predictive validity or usefulness.
There is also convergence regarding the instrumental value of a model as a tool to simplify a complex situation. However what shines through as a distinguishing feature of the systems model is that the system-cum-model can be used to describe some other system or to think about or make sense of it. This functionality of the model as a reasoning tool is a subtlety that is critical to this study. It is particularly suited to the interpretation of ‘modeling with Sketchpad’ envisaged in this study where the dynamic software is used to model the physical derivative concept, not only graphically and symbolically, but also in dynamic numeric and dynamic graphic senses, simultaneously.
Lesh and Doerr (2003:363) take a further step to examine the question of the model: is it inside or outside the mind? On the one hand they consider models to be conceptual systems (which places them in the cognitive domain) that function with the support of powerful tools (which are external elements), or representational systems (external systems) which places them in the exterior (real world) domain, each of which emphasizes and de-emphasizes (or ignores or distorts) somewhat different aspects of the underlying conceptual system (which makes them subjective).
On the other hand, man-made conceptual systems (mental networks, or theoretical frameworks, including mathematical structures and systems) are viewed to be partly embedded in conceptual tools that involve electronic gadgets (computers), specialized symbols (e.g. Sketchpad and calculus symbolization), language, diagrams, organizational systems or experience-based metaphors. What stands out here is the function and role of a ‘tool’ in the modeling process. The use of a tool is an integral part of the act or process of reasoning. Because reasoning is a cognitive process, the tool is therefore not entirely outside the mind, and because the tools can be independently external, it means the reasoning process is not entirely confined to the head. Again these interpretations are in accord with the cognitive science view of a model and the intended use of Sketchpad. In other words, the modeling we envisage with Sketchpad is as a conceptual system with both internal (cognitive) and external dimensions in interaction with one another to solve real-world or quasi real-world problems. In their turn, conceptual tools have both an external and internal existence.

Modeling and representations

In comparing models with representations, Lesh and Doerr (2003:363) note that the meaning of a model (or conceptual system) tends to be distributed across a variety of interacting systems which may involve written symbols, spoken language, pictures or diagrams, concrete manipulatives or experience-based metaphors. These components or forms of representation systems seem identical in description to the characterization of conceptual tools made in 3.4.2 above. The differences are that while on the one hand models emphasize the dynamic and interacting characteristics of systems being modeled, representations draw attention to the objects within these systems. On the other hand, while models refer to functioning whole systems, representations tend to be treated as inert collections of (static) objects to which manipulations (dynamism) and relationships must be added in order to function.
The above characterization of representations seems to be partly compatible with the representational capabilities of Sketchpad. On the one hand the graphic, symbolic and numeric representations in static form seem to fit the characterization of representations quite perfectly. On the other hand the dynamic numeric, and the dynamic graphic representation capabilities could pass the test of being systems in these criteria because they are imbued with motion (dynamism) to illustrate the relationship between variables. Alternatively they can be interpreted as dynamic objects, in which case the restriction to static would need to be revised. When the tangent is animated, the derivative of a function is literally set in motion in Sketchpad, which is a greatly extended visualization or degree of modeling. In other words, under animation, the representational objects transmute to a representational system.
Lesh and Doerr (2000) summarize their comparison of models and representations by stressing that the modeling process involves the interaction among three types of systems:
a) the internal conceptual systems, b) the representational systems which function both as externalizations of internal systems and as internalizations of external systems and c) external systems that are experienced in nature or are man-made artifacts. This summary accurately represents the spirit of modeling envisaged with Sketchpad in this study. The summary diagram however mismatches the verbal summary (See Figure 3.3).
The authors argue that conceptual systems seem to exist in the head, while representational systems are embedded in spoken language, written symbols, pictures, diagrams and concrete models that people use to both express their mental (conceptual) systems and to describe external systems. Van Oers (1996:93) shares a somewhat similar view in suggesting that representational systems are an activity of figuring out, refining their representativity and communication value with others. He further contends that although they are carriers of meaning, they are not themselves the creator of meanings. In other words the cognising agent in interaction with the symbol system creates the meanings.
External systems are considered to be man-made artefacts (e.g. economic systems, communications systems, mechanistic systems, mathematical structures and systems) projected into the world to become part of the experienced world of others. It is further suggested that the boundaries between the systems are fluid, shifting and at times ambiguous (ibid. p. 363). In other words the systems are viewed as partly overlapping, interdependent and interacting and Figure 3.4 can be a plausible alternative diagrammatic representation.

TABLE OF CONTENTS
CHAPTER I: INTRODUCTION AND ORIENTATION 
1.1 Introduction
1.2 Statement of the problem and purpose of the study
1.3 Research methodology adopted in this study
1.4 Definition of key terms
1.5 Significance of the study
1.6 Organization of the thesis
1.7 Conclusion
Chapter II: Historical development of the derivative and its teaching
2.1 Introduction
2.2 A brief history of the derivative
2.3 The traditional approach to the teaching of the derivative
2.4 The Calculus Education Reform Movement and the impact of technology
2.5 Research on the teaching of the derivative in dynamic software environments
2.6 Studies describing how technological tools can be employed in qualitatively different ways
2.7 Insights from the review of the literature
2.8 Initial aspects of a learning trajectory in Sketchpad
2.9 Conclusion
CHAPTER III: MODELING AND CONTEXTUALIZATION IN SKETCHPAD
3.1 Introduction
3.2 Philosophical background to modeling
3.3 An analysis of the meaning of a model 6
3.4 Modeling as a process in Mathematics Education
3.5 A synthesis of the various perspectives on modeling
3.6 Towards an integrated meaning of modeling in this study
3.7 Modeling as a teaching strategy in each cycle
3.8 Conclusion
CHAPTER IV: RESEARCH METHODOLOGY AND THE DESIGN OF INSTRUMENTS
4.1 Introduction
4.2 Research Methodology
4.3 Design of Instruments
4.4 Conclusion
CHAPTER V: DATA AND DISCUSSION OF DATA
5.1 Introduction
5.2 Test investigations
5.3 Sketchpad Activities to enhance students’ conceptual understanding of the graphical representation of the derivative
5.4 Analysis of post task-based interviews
5.5 Posttest results
5.6 Conclusion
CHAPTER VI: SUMMARY AND CONCLUSIONS
6.1 Introduction
6.2 Summary of the findings
6.3 Implications of the results for mathematics learning
6.4 Limitations of the study
6.5 Implications for future research
Bibliography
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