The concept of equations and inequalities have a significant place in mathematics (Bazzini & Tsamir, 2002; 2004). They are intertwined in various mathematical subjects such as algebra, analysis, and linear programming (Bazzini & Tsamir, 2004; Tsamir & Almog, 2000), and many concepts of geometry are based on inequalities (Kaplan & Acil, 2015), in addition, equations, and inequalities provide a complementary perspective to each other (Tsamir & Almog, 2000). Equations and inequalities are involved in problem-solving techniques (Altun et al., 2007), abstraction, and mathematical modeling that simulates real-life situations (Karataş & Güven, 2010). It is recommended that students learn to depict cases involving equations and inequalities and how to solve equivalent expressions, equations, and inequalities by inferring their meaning (NCTM, 2000). However, it is claimed that students’ conceptual representations of the algebraic expression and equation lack completeness and accurateness, students at all levels frequently struggle with the concept of inequality, find it extremely difficult to interpret inequality solutions, and students who make more mistakes in these concepts exhibit low mathematical understanding (Stewart, 2016). Students encounter a variety of challenges when they attempt to solve equations and inequalities, such as an inadequate understanding of the role of the equals and inequality signs (Almog & Ilany, 2012; Blanco & Garrote, 2007; Knuth et al. 2006), a lack of understanding of the symbolic representation of variables and coefficients in an equation (Kilpatrick & Izsak, 2008), and changing the direction of the inequality when multiplying by the negative number (Cortes & Pfaff, 2000). According to Bazzini and Tsamir (2001), students who discovered inequalities with traditional methods (by doing algorithmically memorized steps) had difficulty when presented with non-traditional tasks. Moreover, Tsamir and Bazzini (2004) found that students think that inequalities should result in inequalities/intervals, and solving the inequalities and equations requires the same process. It is suggested that to address these challenges, solution processes of equation and inequality should not be introduced directly and quickly, that the symbols used be clearly differentiated and have meaning for the students, that the differences between the concepts of equation and inequality be made clear, and that every-day, visual-geometric and algebraic language should be used interchangeably (Blanco & Garrote, 2007; Bazzini & Tsamir, 2002).

The existing research emphasizes the difficulties students have with equation and inequality concepts. Most of the studies are based on cognitive perspectives, and some of them explore classroom interaction through sociocultural perspectives. In this study, we focus on the equation and inequalities in the classroom discourse to analyze the understanding of learners and teachers. Our study uses a commognitive perspective because it highlights the interaction in a natural classroom setting and enables us to analyze the exploration of learners and teachers. we adopted a methodological lens that Nachlieli & Tabach (2019) provide for ritual-enabling and exploration-requiring opportunities to learn. We interpreted explorative-requiring opportunities to learn as explorative participation and analyzed data on how (procedure) and when (initiation and closure) explorative participation was actualized. We aim to explore the characterization of explorative participation of prospective mathematics teachers and lecturers on equation and inequality concepts in the context of classroom discourse. We address the following question: How do the characterization of explorative participation of prospective mathematics teachers and lecturers on equation and inequality concepts in the context of classroom discourse?

The data for this study was conducted in the context of a “basic mathematical concepts” course in a mathematics education department in Turkey. We collected data from classroom observations conducted in the context of a “basic mathematical concepts” course for 20 seniors studying at a mathematics education department. The lecturer is a professor with a Ph.D. degree in mathematics and works in the mathematics education department. Prospective mathematics teachers (PMT) take mathematics education content courses (such as calculus, discrete mathematics, and linear algebra), mathematics education courses (such as geometry education, algebra education, material design, and technology in mathematics education), and pedagogical courses (such as developmental psychology, classroom management, approaches and theories of teaching and learning). In the context of a “basic mathematical concepts” course, PMTs analyze and discuss basic mathematical concepts (such as propositions, equations, inequalities, polygons, vectors, functions, and transformation). In this study, we focused on the equations and inequalities concepts. In this course, PMTs work in groups of four. PMTs investigate the origins, meaning, and history of specific mathematical concepts, then analyze and categorize the definitions of the particular concept in the literature. After PMTs examine the equation and inequalities concepts, each group presents one clear mathematical concept in the classroom and comprehensively discusses the definitions. Each group justifies and supports their ideas regarding the definitions of equations and inequalities. Finally, the presenting group provided a final and concise definition of the equations and inequalities they had discussed. Classroom observations collected through a video camera were transcribed into participants’ native language and translated from Turkish into English. The transcripts of the classroom observations included participants’ utterances and their visual mediators and actions. The data were analyzed regarding participants’ and lecturers’ characterization of explorative participation (Sfard, 2008). We adopted a methodological lens that Nachlieli & Tabach (2019) provide for ritual-enabling and exploration-requiring opportunities to learn. We interpreted explorative-requiring opportunities to learn as explorative participation and analyzed data on how (procedure) and when (initiation and closure) explorative participation was actualized.

In this study, we have explored the characterization of explorative participation of prospective mathematics teachers and lecturers on equations and inequalities in the classroom discourse. The main discussion in this classroom is driven by the definitions of the available equations and inequalities found in the literature. The main goal of this study is to analyze the definitions of equations and inequalities that enable PMTs to comprehend the flexibility and logical structure of symbolic representations of algebraic expressions and equations, inequalities signs, and their mathematical meaning. PMTs provide three themes on definitions of inequalities and five themes on definitions of equations. By exploring their definition decisions, the lecturer has orchestrated the classroom discourse. When PMTs were explaining their ideas, the lecturer asked exploratory questions. The lecturer prompted exploratory questions for classroom discussion to obtain exploratory engagement, where the actions aligned with the lecturer's goal and were applied flexibly in a logical structure (Nachlieli & Tabach, 2019). Each group has provided logical justifications for their decision-making process. Prospective mathematics teachers actively participated in the classroom discourse by producing mathematical narratives focused on expected outcomes. We discovered that the lecturer had initiated words such as what, why, find, and frequently explain, allowing PMTs to engage in exploratory practices.

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