A calculus student’s understanding of graphical approach to the derivative through quantitative reasoning

The concept of derivative is used in many areas including applied problems and requiring mathematical modelling in different disciplines. One of the most important approaches for teaching the derivative is to support students in visualizing the concept. Also, it is necessary to shift researchers and teachers’ focuses to students’ dynamic mental actions while learning derivative in order to conduct effective teaching process. With this necessity, I focused on the perspective of quantitative reasoning related to the graphical approach to the derivative. This study aims to reveal a calculus student’s mental actions related to the graphical approach to the derivative. The data were collected from a first-year calculus student engaged in the task requiring graphical interpretation of the derivative. Results showed that the student’s understanding of the slope shaped her inferences about the tangent line because the quantity of ratio is prior knowledge for learning the instantaneous rate of change. Besides, as the student had the idea of correspondence related to the concept of function, she had difficulties in interpreting the global view of the derivate. This result suggests that having global view of the derivative requires a strong understanding of function and rate.

Mobile Computing as a Support to Teach the Concept of Function

We make a proposal to teach the concept of function using mobile computing. This proposal is based on research that has support in education theories such as Constructivism and Problem Solving Learning. In the first part we show the difficulties that undergraduate Computer Science students have in the first semester of Engineering while working with that mathematical concept. And application to be used by the students anywhere and anytime in mobile devices is designed that process data and is intended to show them the concept of function in a problem solving situation. The proposed activities are part of the education methodology used in the research and within this paper we show the diagnostic questionnaire as one of the methodological tools and its results supporting the designed activities and its application to mobile devices

The Role of Representations in the Understanding of Mathematical Concepts in Higher Education: The case of Function for Economics Students

the case of higher education most studies were conducted at pedagogical departments for prospective teachers and mathematical departments. The present study concentrates on university students who attend a course on mathematics as part of a program at the Faculty of Economics and Management. It examines students’ affective and cognitive behavior in solving representation tasks concerning their understanding of exponential and logarithmic functions. Results confirmed the existence of a comprehensive model with significant interrelations among general beliefs, self-efficacy beliefs and cognitive behaviour about the use of representations in general and, in the case of the specific concept. Regression analysis indicated the predominant role the self-efficacy beliefs play in the use of representations in defining the concept of function and solving recognition and translation tasks. Implications about the teaching of mathematics in higher education are discussed.

Student teachers’ knowledge of students’ difficulties with the concept of function

An important part of the mathematics syllabuses at the secondary school level in most countries is the concept of function. However, secondary school students often experience difficulties with this concept. These difficulties are well-known in the research literature. The study applies the mathematical knowledge for teaching (MKT) framework, including the category knowledge of content and students (KCS). Teachers’ ability to anticipate students’ difficulties is one aspect of KCS. The aim of this study is to investigate secondary mathematics student teachers’ KCS regarding the concept of function. Ten mathematics student teachers participating in a Supplementary Teacher Education Program answered a questionnaire about fictive secondary school students’ various difficulties with the concept of function. Follow-up interviews were conducted with four of the respondents. Compared to the findings of previous research on students’ difficulties with the concept of function, the respondents in the study sometimes provide reasonable suggestions about the sources of students’ difficulties. Some of the respondents demonstrate an aspect of KCS when they suggest that students can reason that a function must be defined by one algebraic expression only, and that students only know about continuous functions. However, no respondent suggests that one source of students’ difficulties with a constant function with an implicit domain is the missing domain. In addition, some respondents take for granted that students can interpret the algebraic representation of a piecewise-defined function and translate it into a graph.

How to Teach the Basic Concept of Function in Senior High School: a Lesson Study

Teaching methods are a complex topic in mathematics education. This study aims to analyze the teaching methods of previous relevant studies and design a new lesson study on how to teach the topic of functions at the high school level in China. Lesson study focuses on the basic concepts of function and problem-solving abilities. The researcher uses the research and development method to teach the material function at the high school level. This Lesson Study is used to teach in China. The researcher explains 4 important aspects in designing a lesson study, namely the introduction or opening section, the instructional section or core section, the assessment section, and the closing section.

Mono- and Multi-Representational Learning of the Covariational Aspect of Functional Thinking

Using multiple external representations is advocated for learning in STEM education. This learning approach assumes that multiple external representations promote richer mental representations and a deeper understanding of the concept. In mathematics, the concept of function is a prototypical content area in which multiple representations are used. However, there are hardly any experimental studies investigating the effect of learning functional thinking with multiple representations compared to learning with only one form of representation. Therefore, this article reports on a quasi-experimental intervention study with students from Grade 7, using three measurement time points. The study compared the multi-representational learning of functional thinking with both tables and graphs with mono-representational learning with either tables or graphs. The results show that multi-representational learning led to advantages in learning qualitative functional thinking. However, in quantitative functional thinking, learning with both graphs and tables did not result in higher learning gains than learning exclusively with graphs. Furthermore, students were better able to transfer their knowledge from graphs to tables than vice versa. The results also indicate that multi-representational learning requires more time than mono-representational learning but can lead to higher learning gains. In sum, the results show that the effect of learning with representations is a complex interaction process between learning content and the forms of representation.

La variación física y el uso de las MEA para la comprensión del concepto de función en estudiantes de Ingeniería / Physical variation and the use of MEA for understanding the concept of function in engineering students

En el presente trabajo, se reportan los resultados de una intervención educativa la cual tuvo como propósito analizar la generación del concepto matemático denominado función el cual debe construirse por estudiantes de Ingeniería. Se aborda el tema, utilizando en primera instancia la variación física y posteriormente Actividades Reveladoras del Pensamiento (MEA) por sus siglas en el idioma inglés.