The quadratic function is an important concept for calculus but the students at high school have many difficulties related to this concept. It is important that the teaching of the quadratic function is realized considering the students' thinking. In this context, the aim of this study conducted through a qualitative case study is to reveal the…

This qualitative research aimed to explore the integration of the components of pedagogical content knowledge (PCK) in teaching Linear Equation with one unknown. For the purpose of the study, a single local case study with multiple participants was used. The selection of the participants was made based on various criteria: having more than 5 years…

One area of algebraic thinking essential for students' success is a relational understanding of the equal sign. Research has indicated a positive correlation between students' relational understanding of the equal sign and their equation-solving performance, suggesting that students' early conception of the equal sign may affect their learning and…

This paper investigates the challenges faced by secondary school teachers and pupils in the teaching and learning of algebraic linear equations. The study involved 80 grade 11 pupils and 15 teachers of mathematics, drawn from 4 selected secondary schools in Mufulira district, Zambia in Central Africa. A descriptive survey method was employed to…

Understanding how students learn is important for both researchers and teachers. In terms of mathematics, constructing a new mathematical structure depends on conceptual understanding and connection with previous construct and they needs to be consolidated. A construct can be consolidated when the construct is recognized and used in the further…

The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes…

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

To understand relationships between students' fractional knowledge and algebraic reasoning in the domain of equation writing, an interview study was conducted with 12 secondary school students, 6 students operating with each of 2 different multiplicative concepts. These concepts are based on how students coordinate composite units. Students…

The concept of multiple representations of functions and the ability to make translations among representations are important topics in secondary school mathematics curricula (Moschkovich, Schoenfeld, & Arcavi, 1993; NCTM, 2000). Research related to students in this domain is fruitful, while research related to teachers is underdeveloped. This…

The concept of multiple representations of functions and the ability to make translations among representations are important topics in secondary school mathematics curricula (Moschkovich, Schoenfeld, & Arcavi, 1993; NCTM, 2000). Research related to students in this domain is fruitful, while research related to teachers is underdeveloped. This…

Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit…

To investigate student difficulties in solving word problems in algebra, we carried out a teaching experiment involving 51 Indonesian students (12/13 year-old) who used a digital mathematics environment. The findings were backed up by an interview study, in which eighteen students (13/14 year-old) were involved. The perspective of mathematization,…

While an algebraic expression is typically assigned a regular structure, this article introduces the concept of personal structure ("Strukturierung"); here, the structuring of an algebraic expression is understood as the act of forming relationships between its parts. This concept is used for the analysis of interviews in which experts and novices…

To investigate relationships between students' quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems…

Students' struggles with algebra are well documented. Prior to the introduction of functions, mathematics is typically focused on applying a set of arithmetic operations to compute an answer. The introduction of functions, however, marks the point at which mathematics begins to focus on building up abstractions as a way to solve complex problems.…