Topic Study Group No. 16: Teaching and Learning of Calculus

This paper addresses a practical issue encountered by many lecturers teaching first-year university engineering mathematics. A big proportion of students seems to be able to find correct solutions to test and exam questions using familiar steps and procedures. Yet they lack deep conceptual understanding of the underlying theorems and sometimes have misconceptions. In order to eliminate misconceptions, and for deeper understanding of the concepts involved, the students were given the incorrect mathematical statements and were asked to construct counter examples to prove that the statements were wrong. They had enough knowledge to do that. However, for most of the students that kind of activity was very challenging and created conflict. 127 students from two universities, in Germany and New Zealand, were questioned regarding their attitudes towards the method of using counter examples for eliminating misconceptions and deeper conceptual understanding. The vast majority of the students...

MEDITERRANEAN JOURNAL

Brain-challenging puzzles have attracted people for a very long time. Paradoxes constitute a special type of puzzle aimed to reveal and emphasize an inconsistency or contradiction resulting from some mental experiments in mathematics. Their resolution teaches us to stay alert and be aware of possible flaws of various kinds. Many paradoxes, such as those of Zeno and Russell, greatly influenced the shape of mathematics as we know it today. That suggests a possibility to incorporate the study of paradoxes in standard mathematical courses. But how productive may it be? At which stage of their study will students benefit from being exposed to paradoxes? How one can practically do it in the classroom? This paper is an attempt to address some aspects of these important questions. We discuss the nature and role of paradoxes in the process of understanding, along with potential problems and advantages of their use in study. We give several examples of mathematical paradoxes in both the historical and the classroom context. A short survey results outline an idea of the audience reaction and suggests further directions for research. We conclude that the pedagogical payoff of the use of paradoxes in the classroom is currently underestimated and a consistent study of the impact of paradoxes on learners will allow us to develop a teaching portfolio which takes a comprehensive advantage of the natural curiosity of the mind towards puzzles.

2019

Permission granted by the Australian Association of Mathematics Teachers (AAMT) for use on ResearchOnline@ND.

Educational Studies in Mathematics, 2008

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention.

Uniciencia , 2018

There is a wide diversity of approaches to solving problems in the teaching of mathematics. In particular, the meaning of “problem solving” differs between theory and practice. In the teaching of higher mathematics, problem solving is frequently used in single variable differential and integral calculus, as indicated by course contents and the number of university programs that include it in their curricula. We therefore investigated the ways in which mathematics teachers use problem solving in the teaching of single variable differential and integral calculus. A questionnaire was applied to teachers with experience in teaching single variable differential and integral calculus from the Universidad de Costa Rica, the Universidad Nacional de Costa Rica, the Instituto Tecnológico Costa Rica, and the Universidad Estatal a Distancia. The results reveal contradictions between teachers’ conceptions of what a mathematical problem is and their implementation of problem solving in the classroom.

Educational Studies in Mathematics, 1990

Teaching and Teacher Education, 1991

This is a third report on a naturalistic study in Israel of the role mathematical paradoxes can play in the preservice education of high school mathematics teachers. The study examined the potential of paradoxes as a vehicle for: (a) sharpening student-teachers' mathematical concepts; and (b) raising their pedagogical awareness of the constructive role of fallacious reasoning in the development of mathematical knowledge. Course material development and data collection procedures are described. Results obtained through written responses and class-video-tapes are analyzed and discussed. The findings indicate that the model of dealing with paradoxes as applied in this study has relevance to such aspects of mathematics education as cognitive conflicts, motivation, misconceptions, and constructive learning. Generalizations are discussed.

ACTA DIDACTICA NAPOCENSIA, 2012

This paper is going to analyse errors and misconceptions in an undergraduate course in Calculus. The study will be based on a group of 10 BEd. Mathematics students at Great Zimbabwe University. Data is gathered through use of two exercises on Calculus 1&2.The analysis of the results from the tests showed that a majority of the errors were due to knowledge gaps in basic algebra. We also noted that errors and misconceptions in calculus were related to learners' lack of advanced mathematical thinking since concepts in calculus are intertwined. Also in this study we highlight some common errors/ mistakes which can be done by lecturers during the teaching process.Students studying calculus often make the same mistakes and similarly lecturers teaching calculus have patterns of mistakes. This paper is derived from practical situations hence it is open to updating and can be adapted by other calculus teachers in different setups.

2013

The study reported herein is part of a larger studythat examined high-school students' understanding of the roles of examples in proving. Data is based on a series of students' interactions with specially designed mathematical tasks that elicit their thinking. The findings provide a complex account of students' conceptions and reveal inconsistences in their understanding. In particular, all students in our study exhibited indicators of understanding that for a universal statement to be true it has to hold for all cases. At the same time, some of these students remained convinced that a statement can be 'proven' through examination of several confirming examples.

2014

We report here on a study of the opportunities for creative reasoning afforded to first year undergraduate students. This work uses the framework developed by Lithner (2008) which distinguishes between imitative reasoning (which is related to rote learning and mimicry of algorithms) and creative reasoning (which involves plausible mathematically-founded arguments). The analysis involves the examination of notes, assignments and examinations used in first year calculus courses in DCU and NUI Maynooth with the view to classifying the types of reasoning expected of students. As well as describing our use of Lithner’s framework, we discuss its suitability as a tool for classifying reasoning opportunities in undergraduate mathematics course