In this report, we present cases where students constructed new quantities through operating on quantities that does not fit the definitions of existing theories on quantitative operations. As a result, we identified five quantitative operators--operators that can be used on single qualities in order to transform the quantity to a new…

In this paper, I propose a new construct named "analytic equation sense" to conceptually model a desired way of reasoning that involves students' algebraic manipulations and use of equivalent expressions. Building from the analysis of two existing models in the field, I argue for the need for a new model and use empirical evidence to…

This study was performed to determine preservice elementary mathematics teachers' solution performance of systems of linear equations using geometric representation, algebraic representation and transition from geometric representation to the algebraic representation and the predictive role of preservice elementary mathematics teachers'…

In this chapter we are interested in how undergraduate physics students in three countries experience the equations they meet in their education. We asked over 350 students in the USA, Australia and Sweden the same simple question: How do you know when you understand a physics equation? Students wrote free-text answers to this question, and these…

Research has shown that tape diagrams are beneficial for algebra learning. However, it is unclear whether certain visual features of tape diagrams have implications for learning. We investigated, with undergraduate students and math teachers, whether tape diagrams with different visual features (color, presence of outer lines, and position of the…

This research reports on the teacher language and gesture that contributed to shifts in thinking about the equal sign and equations observed in twenty kindergarteners who took part in an early algebra intervention. Our analysis revealed ways in which the teacher used language and gesture to support students in moving from describing and working…

The appropriateness of guess and check as a problem-solving strategy has been debated. This qualitative study examines the use of guess and check by middle-grades students to solve linear systems of equations. Students' reasoning is examined within the number sequences framework, which is based in part on students' units coordination. Only…

It is commonly known that mathematics is used in many subjects in physics and chemistry courses and the relations between mathematics and variables are more easily understood. These courses also comprise a number of equations that correspond to a mathematical correlation and are also called formulas. Therefore, the aim of the present study is to…

Literature typically describes mathematization, the process of transforming a real-world situation into a mathematical model, in terms of desirable actions and behaviors students exhibit. We attended to STEM undergraduate students' quantitative reasoning as they derived equations. Analysis of the meanings they held for arithmetic operations (+, -,…

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and…

Student focusing and noticing, which drive reasoning, are important but under researched aspects of student learning. Quadratic functions representations are perceptually and conceptually complex and thus, offer much for students to focus on and notice. Our study compared a teacher's goals for student focusing and noticing during quadratic…

This paper provides evidence that a short, fully online, well-constructed diagnostic test based on research literature can give teachers information about their students' thinking and strategies that is sufficiently accurate to use for formative assessment purposes. The example is a test for students beginning to learn to solve equations. The main…

Mathematics education scholars have generally classified students' conception of the equal sign as either operational or relational. Adding to these conceptions, Jones (2008) introduced the idea of substitutional conception. Building off these scholars, I introduce a form of understanding the equal sign that includes a transformative equivalence…

The concept of equivalence can be elusive to students and can be confounded with unproductive understandings of the equals sign. Using the game-based digital algebraic notation system, From Here to There! (FH2T), students explore ideas of equivalence by dynamically transforming expressions or equations among mathematically equivalent states. In…

Systems of equations are an important topic in school mathematics. However, there is limited research examining productive ways of supporting students' understandings of systems of equations. In this paper, we first present a conceptual analysis of potential ways students may leverage their quantitative and covariational reasoning to graph systems…