Teaching Of Calculus Research Articles

Using an advanced graphing calculator in the teaching and learning of calculus

The purpose of this study was to investigate how the use of TI-Nspire™ could enhance the teaching and learning of calculus. A conceptual framework for the use of TI-Nspire™ for learning calculus in a mathematics classroom is proposed that describes the interactions among the students, TI-Nspire™, and the learning tasks, and how they lead to the learning of calculus. A design experiment was conducted in a class of 35 students from a secondary school in Singapore. Use of TI-Nspire™ was integrated into the teaching and learning of calculus concepts in the classroom with the aid of TI-Nspire™ Navigator, a wireless classroom network system that enables instant and active interaction between students and teachers. It was found that the appropriate use of graphical, numerical and algebraic representations of calculus concepts using TI-Nspire™ enabled students to better visualize the concepts and make generalizations about relevant mathematical properties. In addition, the students were able to link multiple representations, especially algebraic and graphical representations, to improve their conceptual understanding and problem-solving skills. Six roles of TI-Nspire™ in classroom mathematical practice were identified from the findings of the experiment; TI-Nspire™ was used as an exploratory tool, graphing tool, confirmatory tool, problem-solving tool, visualization tool and calculation tool. This suggests that TI-Nspire™ is a multi-dimensional tool that supports mathematics learning. Overall, the findings of the study indicate that TI-Nspire™ is an effective tool to develop mathematical concepts and promote learning and problem solving.

Les questions de développement curriculaire à travers un exemple: l'enseigment de l'analyse en France au lycée depuis le debut du XXème siècle (Republicação)

Curricular evolutions obey complex dynamics. These are shaped by conditions and constraints situated at very different levels and involve a diversity of actors. In this article, we approach this complexity, taking as an example the case of the teaching of Calculus in highschool in France from the beginning of the twentieth century, and using as a theoretical framework the anthropological theory of didactics (ATD) developed by Chevallard. We first introduce the constructs of ATD we mainly refer to: the idea of praxeology and the hierarchy of levels of co-determination, then we trace curricular evolutions in the highschool teaching of Calculus in France distinguishing seven different periods since the 1902 reform. Along the analysis, we try to identify some characteristics of this dynamics and associated didactical phenomena possibly transcending this particular case.

How high is the tramping track? Mathematising and applying in a calculus model-eliciting activity

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required them to find a graphical representation of an anti-derivative of a function. When determining the value of the anti-derivative as a measure of height, they mathematised the situation to develop a mathematical model, and attempted to apply their knowledge of integration that they had previously learned in class. However, the participants favoured their more primitive mathematised knowledge over the formal knowledge they tried to apply. We use these results to argue for calculus instruction to include more modelling activities that promote mathematising rather than the application of knowledge.

Notations pour le calcul réseau

Network calculus involves many notions that are both due to the applications concerned (flow, server, delay...) and to the model (arrival curves, service curves...). The mathematical framework is based on the (min,plus) algebra. This article attempts to provide some notations associated to these notions. The aim is two fold: on the one hand, simplification of the manipulation (model, proofs) and appropriation (diffusion, teaching) of network calculus and in the second hand, formalization and clarification of some notions (notion of server for example).

Mathematics textbooks and their potential role in supporting misconceptions

As a fundamental resource, textbooks shape the way we teach and learn mathematics. Based on examination of secondary school and university textbooks, we describe to what extent, and how, the presentation of mathematics material–in our case study, the concept of the line tangent to the graph of a function–could contribute to creation and strengthening of students’ misconceptions. Our findings, roughly classified in several categories, raise awareness of non-obvious problems that need to be addressed when teaching calculus.

Applications GeoGebra into teaching some topics of mathematics at the college level

This paper presents new trends in technology and learning through GeoGebra, which could be especially important for the future development of e-learning for College mathematics. Also, contribution of this paper is presentation of methodological frames on several specifics examples for teaching mathematics at the college level on interactive and creative way. Discussed in this paper are the findings of a didactic experiment where some number of undergraduate students were exposed to an innovative (to the departments) method of teaching and learning a part of calculus supported by GeoGebra. Statistical analysis confirmed the fact that the use of the applets created with the help of GeoGebra and used in differential calculus teaching had a positive effect on the understanding and knowledge of the students.

A TRAILER, A SHOTGUN, AND A THEOREM OF PYTHAGORAS

Counselor: Please tell the Court your name. Expert Witness: My name is Will Kazez Counselor: No, no, no! Your name is… This is not a good start. I am not naturally a nervous person. I have survived teaching calculus to a large class that included the entire freshman football team of the University of Pennsylvania, but I've never been an Expert Witness. Even though I'm confident of the mathematics, I'm not sure I like the idea of being cross-examined. But still, I'm just rehearsing my testimony with the lawyer, and even if I've got my own name a little wrong, what's the worry? At any rate, lawyers do not like being interrupted.

Study on the Intuitive Introduction in Calculus Teaching

In the popularization course of higher education, students entering into the higher colleges are not those excellent students with excellent achievements in the senior school, so the teaching of college mathematics in higher colleges should transform from the learning emphasis to popularization emphasis for traditional excellent higher education. In the transformation process, as a main basic course for the students of the engineering course, the calculus has many problems in the teaching. Based on many years’ teaching practice and theoretic research, in this article, we first analyze the problems and antinomies faced by calculus teaching, demonstrate the feasibility of intuitive introduction method from the mathematical essential and the rule of human thinking, and finally explain the implementation method of the intuitive introduction combining with concrete teaching cases.

The Time Invariance Principle, the absence of ecological chaos, and a fundamental pitfall of discrete modeling

This paper is to show that most discrete models used for population dynamics in ecology are inherently pathological that their predications cannot be independently verified by experiments because they violate a fundamental principle of physics. The result is used to tackle an on-going controversy regarding ecological chaos. Another implication of the result is that all dynamical systems must be modeled by differential equations. As a result it suggests that researches based on discrete modeling must be closely scrutinized and the teaching of calculus and differential equations must be emphasized for students of biology.

Transcendental Functions and Initial Value Problems: A Different Approach to Calculus II

Byungchul Cha (cha@muhlenberg.edu) teaches at Muhlenberg College in Allentown, Pennsylvania, after working for three years at Hendrix College. He received his B.S. from the Korea Advanced Institute of Science and Technology, and his Ph.D. from Johns Hopkins University. His area of specialization is number theory. When he is not teaching calculus, he enjoys engaging his young son in doing other mathematics, such as counting from 1 to 10 and drawing octagons.

Cognitive Computer Tools in the Teaching and Learning of Undergraduate Calculus

The purpose of this study was to explore the use of a cognitive computer tool by undergraduate calculus students as they worked cooperatively on mathematical tasks. Specific attention was given to levels of cognitive demand in which the students were engaged as they completed in-class labs with the assistance of MathCAD. Participants were assigned to eight heterogeneous working groups consisting of four students each. One group was chosen as the focus of the case study. Data included student questionnaires, individual interviews, assignments, audio transcriptions of student discussions, and video recordings. Open and axial coding was used to analyze the data. Participants believed that the use of MathCAD allowed them to explore mathematics, spend more time on interpreting results, and focus on understanding. The cognitive computer tool reduced the reliance on low-level thinking skills and allowed for creativity in problem solving, permitting students to move toward high levels of thinking.

Una investigación del aprendizaje del cálculo aritmético

The aim of this article is that of showing a specific approach in the Didactics of Arithmetic Calculus, and the results that have been achieved using it with about 300 students in the first year of SUP: we also want to show certain criteria which could allow us to improve our method. Such criteria are based both on the realization of the didactIc method itself and on the assumption of certain cognitive factors which have great influence on the learnin& of Arithmetic Calculus. Throughout the developing of our approach in the Didactics 5f Arithmetic Calculus we have emphasized the sections devoted to the teaching of calculus, error enalysis and calculus problem solving (according to Polya).

The Rule of Four in Statistics

The Rule of Four in Statistics Volume/Issue Vol 6, Issue 3 Date Tuesday, 1 August, 2006 Journal Name MSOR Connections Author(s) Warren Gilchrist Abstract Over recent years there has been a major reform movement in the USA focussed on the teaching of calculus in colleges. Part of this development is an approach to teaching mathematical ideas termed the Rule of Four. The approach is totally in tune with the UK approaches to teaching statistics developed over recent years. Discussions at the RSS Centre for Statistical Education have led to a statistically oriented version of this Rule. The purpose of this article is to define and illustrate this Statistical Rule of Four. Any statistics teacher reading this will be well familiar with the four elements of this Rule. So in a sense this article is redundant. However the issue raised here is not do we know the ideas, but do we put them together in practice? This short article is thus a reminder. The box below gives the basic statement of the Rule: DOI doi:10.11120/msor.2006.06030013 File upload msor.6.3m.pdf [1] Taxonomy: work area

THE CURL OF A VECTOR FIELD: BEYOND THE FORMULA

ABSTRACT It has been widely acknowledged that there is some discrepancy in the teaching of vector calculus in mathematics courses and other applied fields. The curl of a vector field is one topic many students can calculate without understanding its significance. In this paper, we explain the origin of the curl after presenting the standard mathematical formulas. We investigate when and why a vector field yields an in-spot spin, also known as curl, and develop intuition to predict the sign of the curl of a vector field without calculating it. As an application of the curl, Stokes' theorem and its physical interpretation are presented with simple illustrations.

Should we reform the teaching of calculus?

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Teaching and learning conceptions in Engineering Education: an innovative approach on Mathematics

A worldwide problem in Engineering education is the high rates of students' failure and drop out particularly at the beginning of the course. This is related to the process by which students learn Mathematics. An innovative methodology of teaching calculus was developed and it is presented in this paper. The approach, based on both course modularisation and the error theory, is student centred. It fosters student/lecturer commitment and emphasizes continuous assessment of the process. Knowledge, in this work, is regarded as a construction and reconstruction process based on phenomenological and hermeneutic concepts. Survey results demonstrated the effectiveness of the applied methodology.

Understanding calculus

A number of significant changes have have occurred recently that give us a golden opportunity to review the teaching of calculus. The most obvious is the arrival of the microcomputer in the mathematics classroom, allowing graphic demonstrations and individual investigations into the mathematical ideas. But equally potent are new insights into mathematics and mathematics education that suggest new ways of approaching the subject. In this article I shall consider some of the difficulties encountered studying the calculus and outline a viable alternative approach suitable for specialist and non-specialist mathematics students alike.

Accounting Real Interest

There isn't really anything new regarding interest rates or interest costs, but so many people seem to get confused or worse confounded. Money has two dimensions not just one; not just quantity but also time. Almost any financial/economical activity deals with amounts of money at different points in time. Consequently, such activities are preposterous if the time dimension of money is not correctly processed. Besides the sizes of the amounts, the dates they become payable are especially of paramount importance. In practice as well as in theory, amounts of money often are not calculated as they should be. Even in big international organisations, blue chips at the Stock Exchange, with impressive staffing, things are going badly, as is testified by their own annual accounts. At many schools and universities world-wide, the teaching of financial calculus can, and should be improved. The mean of amounts of money at different dates or whatever 'calculation' which does not give justice to the 'present value' of money, is artificial. Balance sheets stating long term liabilities at face value are fiscally and otherwise acknowledged as 'de jure', but 'de facto' economically speaking, those balance sheets are beyond reality.

Integração entre o ensino de cálculo e o de pesquisa operacional

Pretende-se mostrar neste trabalho como é possível ensinar, de forma integrada, as disciplinas de Cálculo Diferencial e de Pesquisa Operacional. Para tal são necessárias pequenas mudanças na forma como o Cálculo é tradicionalmente ensinado, mas grandes mudanças no ensino tradicional de Pesquisa Operacional. Com estas mudanças, Pesquisa Operacional pode cumprir o papel de disciplina integradora, fazendo a transição entre os ciclos básico e profissional no curso de Engenharia de Produção.

A Preliminary Study of Achievement, Attitudes toward Success in Mathematics, and Mathematics Anxiety with Technology-Based Instruction in Brief Calculus

This study was designed to compare achievement, attitudes toward success in mathematics, and mathematics anxiety of college students taught brief calculus using a graphic calculator, with the achievement and attitudes and anxiety of students taught using the computer algebra system Maple, using a technology based text book. 50 men and 50 women, students in three classes at a large public university in the southwestern United States, participated. Students' achievement in brief calculus was measured by performance on a teacher-made achievement test given at the end of the study. Analysis of variance showed no significant difference in achievement between the groups. To measure change in attitudes and anxiety, responses to paper-and-pencil inventories indicated significant differences in favor of students using the computer.