A group of eighth-graders was presented with a two-day lab exploring graph theory as an enrichment experience. With the school's winter break looming, students were weary of solving linear equations, and this topic was intended to inject some new life into the classroom. In addition to learning about a completely new topic, they would be exposed…

Although there is much research exploring students' covariational reasoning, there is less research exploring the ways students can leverage such reasoning to coordinate more than two quantities. In this paper, we describe a system of covariational relationships as a comprehensive image of how two varying quantities, having the same attribute…

Emergent graphical shape thinking (Moore & Thompson, 2015) is a way of reasoning that is critical across numerous STEM fields. However, evidence indicates that the underlying component ideas for emergent thinking are underdeveloped in school mathematics education (e.g., Thompson & Carlson, 2017), and few studies directly report on…

The purpose of the present research is defining the direction and level of the relationship between 8th grade students' translating among multiple representations skills and their algebraic reasoning and revealing the predictive power on algebraic reasoning. The research was conducted in accordance with relational survey model, which is a…

In this study, based on the analysis of a teaching experiment with middle school students, we propose a framework for describing meanings of a point represented on a plane in terms of multiplicative objects in the context of graphing. We classify those meanings as representing: (1) non-multiplicative objects; (2) quantitative multiplicative…

Middle school is a critical time when students begin formal study of functional relationships in algebra. However, many students struggle in understanding functions as relationships between quantities that change according to a dependency relationship. We report on the influence of Scaling Continuous Covariation in fostering productive ideas about…

This research was motivated by several studies which show that students have difficulties in understanding the material ratio, wherein one contributing factor is the lack of ability in proportional reasoning. The purpose of this study is to produce a trajectory of learning to help students develop abilities in comparison reasoning. Solution…

Implementing an integrated sequence of concrete-representational-abstract depictions of mathematics concepts (CRA-I) can improve the mathematics achievement of students with disabilities, and explicit instructional strategies involving problem-solving heuristics and student verbalizations can help facilitate students' conceptual understanding of…

To prepare students for participation in our society, where interpreting, representing, and manipulating of dynamic phenomena are becoming key activities, we believe that one should start developing a mathematical understanding of change at an early age. We therefore started a design research project to teach the concept of instantaneous speed in…

A renewed emphasis was placed on ratio and proportional reasoning in the middle grades in the Common Core State Standards for Mathematics (CCSSM). The expectation for students includes the ability to not only compute and then compare and interpret the results of computations in context but also interpret ratios and proportions as they are…

Statistics is a domain that is taught in Mathematics in all school levels. We suggest a potential in using an interdisciplinary approach with this concept. Thus the development of the understanding of a situation might mean to use both mathematical and statistical reasoning. In this paper, we present two case studies where two middle school…

No person, place, or thing can capture the attention of a class of sixth graders like "man's best friend." To prompt students' interest in a series of lessons on proportional relationships, the authors brought in a unique teaching aid--a dog. A family dog was used to supply the measurements for scatter plots and variables so that students could…

An nth-term problem involves a sequence. Students must determine which expression will allow them to calculate the nth position of the sequence. To solve such problems, students are to find "a rule that determines the number of elements in a step from the step number." These types of problems help students develop concepts of functions, variables,…

Evolving technology has played an important part in a common quadratic-function lesson. Having been mentioned repeatedly in numerous reform documents, a recurring lesson has involved changing the parameters in f(x) = ax[superscript 2] + bx + c and studying the effects on the graph. In both NCTM Yearbooks and NCTM Standards documents, technology is…

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different…