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We use Action-Process-Object-Schema theory (APOS) to study the development of the differential calculus Schema for two-variable functions. This allows us to obtain information about students' constructions and also gives us information about the notion of Schema. We performed semi-structured interviews with a group of eleven students that had compl...
The purpose of this study is to examine students' meanings for the derivative at a point. While students may associate rate of change with derivative, this does not mean that the meaning they have for derivative is productive. This study explores students' responses to a typical calculus 1 problem that uses derivative to determine a linear approxim...
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... A form of deep learning. Structural disciplines that can be used in organizing the curriculum to engage students in these basic ideas and research methods also provide means for conveying other ideas, topics, and problems to and from school life [25,27]. ...
The model is based on the use of interactive teaching methods. A characteristic feature of the use of interactive technologies is the organization of training that takes into account the inclusion of all the students of a group without exception in the learning process. Joint activity means that each participant makes his or her own individual contribution, whereby in the course of work there is an exchange of knowledge, ideas, and methods of activity. An environment of educational communication is created that is characterized by openness, interaction of participants, equality of their arguments, accumulation of common knowledge, and the possibility of mutual evaluation and control. The use of neural networks to study and predict educational assets will provide research and development organizations and teams with innovative and effective ways of conducting research in the field of educational theory, modeling of the cognitive processes related to formation of different student competencies, and devising more appropriate methods for estimating student educational outcomes.
... The instrumental approach acknowledges the unexpected complexity of technology integration into the mathematics classroom and proposes that the use of ICT tools involves instrumental genesis, a process during which the object or artefact is turned into an instrument (Drijvers et al, 2010). In this paper Sketchpad is the mathematical work tool by which students can be empowered to enhance their conceptual understanding of the derivative as demonstrated in Ndlovu (2008) and Ndlovu et al. (2011). Essentially the success of the concept building process depends on the software affordances 1 , students' knowledge in the form of utilisation schemes (usage schemes for hardware operation and instrumented action schemes for mathematical concept development), and the teacher's instrumental orchestrations (didactical configurations for classroom arrangements, exploitation modes for teaching methods, and didactical performances for teacher interventions during learning). ...
In this paper I illustrate the representational capabilities of Sketchpad that have the potential to enhance a deeper understanding of the derivative concept in introductory calculus if appropriate learning trajectories are designed. Sketchpad is dynamic mathematics software with Trouche's instrumental theory affordances that can support multiple representations of mathematical concepts. The proliferation of digital technologies, under which dynamic mathematics software falls, challenges mathematics educators and teacher educators to accelerate the integration of these new tools into the classroom. To this end I present a hypothesized learning trajectory of the derivative for the instrumental geneses of the derivative as an instantaneous rate of change and as a rate of change function. Six forms of representation of the derivative emerge as a potential part of the mathematics teacher's Technological Pedagogical Content Knowledge (TPACK). A recommendation is made to vigorously equip and capacitate pre-service and in-service mathematics teachers or risk them becoming an impediment. Introduction The integration of new Information and Communication Technology (ICT) tools in the teaching of mathematics and science is actively encouraged worldwide, more so on the back of rapidly expanding digital technology penetration rates even in developing country contexts. The purpose of this paper is to demonstrate the affordances (enablements, potentialities and constraints) of dynamic mathematics software in representing the derivative concept in introductory calculus. The ideas of a model and modelling in mathematics education are examined first and the potentialities of Sketchpad dynamic geometry software in modelling the derivative are explored from an instrumental/documentational genesis perspective before finally locating them in the technological pedagogical content knowledge expected of teachers of mathematics in a technology rich classroom. Modelling in mathematics education 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) are formed (Lesh & Doerr, 2000, p. 362). A model in this sense has a structure made up of components and relationships that connect the component elements to represent a physical reality. In the case of the derivative, in mathematics, the graphical representation yields two conceptualisations or models of a) the derivative as an instantaneous rate of change and b) the derivative as a rate of change function. Each of these models or encapsulations has its own elements that interact to make a representative system. As the gradient at a point, the components include the graph of the functional relationship and a tangent drawn at a point as Figure 1 shows. In contrast to the paper-and-pencil environment, the tangent itself can be viewed as a subsystem constructed through plotting two points (say A and B) on the graph, joining them with a line and animating them to coincide with each other to produce the tangent line.
... That these software packages are relatively new tools makes their efficient integration into mathematics classrooms a pedagogical research imperative. Based on Ndlovu (2008), this article describes a possible theoretical framework for a Sketchpad-mediated teaching sequence from an instrumental approach. The research questions that guided the study are then presented and the research methodology and procedure explained before presenting and discussing the results. ...
... The learning trajectory hypothesised was the culmination of an analysis of the historical development of the derivative in the context of motion (e.g. Zeno's paradoxes, speed and time) and of drawing tangents and finding rates of change or instantaneous speed (see Ndlovu, 2008;Ndlovu et al., 2010). ...
... This research is based on a teaching experiment conducted by the first author involving 20 undergraduate non-mathematics science major students at a university in Zimbabwe (Ndlovu, 2008). The students volunteered to participate in the study, were informed about its purpose and consented to participation. ...
Encouragement to integrate information and communication technologies into mathematics
education curricula is an increasingly universal phenomenon. As a contribution to the
discourse, this article discusses the potential use in the classroom of The Geometer’s
Sketchpad® (Key Curriculum Press, Emeryville, CA, United States) mathematics software in
modelling the derivative and related concepts in introductory calculus. In an empirical study
involving first-year non-mathematics major undergraduate science students, a hypothetical
learning trajectory (HLT) was conjectured and implemented for students to experience the
visualisation and multiple representations of calculus concepts on the Cartesian plane with
a computer graphic interface. The utilisation scheme is interpreted through the lens of the
instrumental1 approach proposed by Trouche. The HLT was partly informed by the historical
development of the derivative as synthesised from the literature on the history of calculus and
partly by the affordances, enablements, constraints and potentialities of Sketchpad itself. The
findings of the study suggest that when exposed to the capabilities of this software, learners
can experience Geometer’s Sketchpad® as an effective visualisation tool or instrument for the
representation and learning of the derivative and related concepts in introductory calculus.
However, the effectiveness of this tool is not a given or a foregone conclusion − it is a product
of the teacher’s instrumental orchestration, gradual learner mastery of the software syntax and
careful resolution of theoretical-computational conflicts that can arise during early use of the
instrument.
In the paper we discuss the pedagogical role that the dynamic mathematics properties of Geometer’s Sketchpad can play in modelling and simulating the derivative and related notions in introductory calculus within a hypothesized learning trajectory. The proliferation of information and communication technologies challenges mathematics educators to engage in experimental research in the wide range of possibilities of enhancing the teaching and learning of mathematics in general and calculus, in particular. In this connection Cuoco (2002) notes that the proliferation opens up a whole new set of mathematical possibilities for students and educators more so when a new tool is designed to serve one field (mathematics) but is used in another (mathematics education).
The hypothesized trajectory was inspired and informed by the historical development of the derivative as gleaned from the literature on the history of the calculus. The models and modelling approach proposed by Lesh and Doerr (2000) influenced the choice and sequencing of model-eliciting activities to develop students’ qualitative understanding of the concept. In an empirical study carried out by the first author (Ndlovu, 2008) involving first year non-mathematics major undergraduate science students at a university in Zimbabwe, six forms of representation enabled by Sketchpad were identified. In this paper we describe the sequence of Sketchpad activities, the nature of the six representations and possible ways in which students can translate from one form of representation to the other. The study suggests that multiple representations can enhance mathematizing and conceptual understanding in learners. The study also reaffirms that the pedagogical value of technology does not lie in the technology itself but more critically in the ingenuity and innovativeness of the teacher in instrumenting it.
