Many students do not have a deep understanding of slope. This paper defines what a deep understanding of slope is in terms of mathematics-education theory. The various factors which help explain why such a deep understanding is difficult to acquire are then discussed. These factors include the following: the different representations for slope;…

The fields of quantitative and covariational reasoning boast a wide range of powerful theoretical tools, which are described carefully in the literature. Less frequent and explicit attention, however, has been paid to writing down detailed, practical guidance for operationalizing these theoretical constructs. Some guidance is provided by…

Helping students reach a clear understanding of the cause-and-effect relationship between changes in parameter and the graph of an equation is the focus of the activity outlined in this article. The behavior of phase shifts has been regarded as counterintuitive for many people, and often, because of this, conflict between student intuition and…

The quality of the intra-mathematical connections made by Mexican university students when solving tasks on the derivative was characterized. The typology of mathematical connections and the quality levels of mathematical connections were used. Interviews were conducted based on eight tasks applied to three case studies. The results showed that…

The authors described the influence of the GeoGebra based environment on students' achievements in the field of functions with parameters, during collaborative learning. Functions with parameters are important because of their application in science. Students had to choose different values for parameters and to examine the properties of the…

This paper offers instructional interventions designed to support undergraduate math students' understanding of two forms of representations of Calculus concepts, mathematical language and graphs. We first discuss issues in students' understanding of mathematical language and graphs related to Calculus concepts. Then, we describe tasks, which are…

This study investigates prospective elementary and secondary school mathematics teachers' ways of reasoning about differentiability at a point and corner points while working on a mathematical modelling activity. Adopting a multiple-case study design, the participants of the study were 68 prospective elementary school mathematics teachers enrolled…

The literature emphasizes the interest and need to learn the concept of continuity with understanding, although this meaning it is not consensual nor specified how it may emerge and be developed in the learning process. In this paper we report the results of a study aiming to analyse Brazilian preservice mathematics teachers' understanding of the…

This is an exploratory study considering instructional methods in mathematics education. The study examined misconceptions arising from selected assignments undertaken by sixty-five students. The students were attending a calculus mathematical course unit on limits of functions, in different academic areas offered at selected tertiary institutions…

In this article we present the results of an investigation into semiotic aspects of an introductory differential equations class from two countries. In order to perform a comparative analysis, students in the USA and Tunisia were given common questions on their midterm exams that examined analytical and graphical competencies. Our results indicate…

Discovery learning has long been a part of mathematics teaching in the elementary and middle grades. Since the 1960s and 1970s, based on the work of Jean Piaget, Jerome Bruner, and others, helping students 'discover' or 'construct' their own understandings of mathematical concepts through well-designed activities facilitated by a competent teacher…

The purpose of this study is to examine the characteristics of students' thinking about graphs while evaluating statements from Calculus. We conducted clinical interviews in which undergraduate students evaluated mathematical statements using graphs to explain their reasoning. We report our classification of students' thinking about aspects of…

This study focuses on students' understanding of multiple representations of functions. It examines student responses to a task in which calculus students are asked to evaluate the derivative at a point of the cubing function when represented piecewise. Results suggest that attending to the graph of the piecewise function does not improve…

In this article, a numerical method integrated with statistical data simulation technique is introduced to solve a nonlinear system of ordinary differential equations with multiple random variable coefficients. The utilization of Monte Carlo simulation with central divided difference formula of finite difference (FD) method is repeated n times to…

This paper reports the results of the experimentation of a didactic engineering for the treatment of the sense of variation of functions with pre-university students. The theoretical references of the investigation are grounded in the theory of didactic situations and the methodological elements in the didactic engineering, the use of…