This paper studies grade 7 (age 13) students' expressed understanding about the equal sign/notion of equivalence in order to investigate what aspects of the concept that could be seen as an established knowledge at lower secondary school/middle school. Using items from different instruments and combining these to a new one that covered a broad…

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to…

This paper investigates and considers factors that affect success in solving a stand-alone geometrical problem by 182 students of the 7th and 8th grades of elementary school. The starting point for consideration is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura), utilising elementary-level geometry…

To efficiently solve mathematical expressions and equations, students need to notice the systemic structure of mathematical expressions (e.g., inverse relation between 3 and 3 in 3 + 5 - 3). We examined how symbols--specifically variables versus numbers--and students' algebraic knowledge impacted seventh graders' problem-solving strategies and use…

The equal symbol has been used in diverse mathematical frameworks, such as arithmetic, algebra, trigonometry, set theory, and so on. In mathematical terms, the equal sign has been used in fixed command of standings. The study reports on the students meaning and interpretations of the equal sign. The study involved Grade 6, 7, and 8 students in a…

All mathematics is comprised of three elements: symbols, nominalizations, and images, and the combination of these three elements determine the mathematical meaning. However, adolescents may not be aware of these three elements. When asked to describe mathematical language, they will often name equations and images, two of the three elements. Over…

With the understanding of importance of pedagogical content knowledge, professional noticing has become one of the research subjects of the mathematics educators for the last 10 years. This qualitative study which explores secondary school students' solution processes regarding the concept of equation ,one of the main components of algebraic…

The aim of the study was to determine and compare 7th, 8th, and 9th grades students' level of use and interpret the literal symbols. In addition, students' responses to questions that require use of different roles of literal symbol were examined to identify the errors. For this purpose, Chelsea Diagnostics Algebra test developed by The Concepts…

Learning algebra is difficult for many students in part because of an emphasis on the memorization of abstract rules. Algebraic reasoners across expertise levels often rely on perceptual-motor strategies to make these rules meaningful and memorable. However, in many cases, rules are provided as patterns to be memorized verbally, with little overt…

Mathematics is rich in visual representations. Such visual representations are the means by which mathematical patterns "are recorded and analyzed." With respect to "vocabulary" and "symbols," numerous educators have focused on issues inherent in the language of mathematics that influence students' success with mathematics communication.…

The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with…

When teachers provide students with every opportunity to visually understand a mathematical concept, process, definition, or notation, they allow them to construct and manipulate relevant and useful images in their minds. Visual understanding has a dynamic character that they do not easily develop with alphanumeric symbols alone despite the…

In many ways, algebra can be as challenging for teachers as it is for students. With so much emphasis placed on procedural knowledge and the manipulations of variables and symbols, it can be easy to lose sight of the key ideas that underlie algebraic thinking and the relevance algebra has to the real world. In the The Xs and Whys of Algebra: Key…

This study examined how literal symbols affect students' understanding of algebraic expressions. Middle school students (N = 322) were randomly assigned to 1 of 3 conditions in which they were asked to interpret an expression (e.g., 4c + 3b) in a story problem. Each literal symbol represented the price of an item. In the c-and-b condition, the…

Learning about linear relationship with one unknown, middle school students may often demonstrate a certain degree of proficiency manipulating algebraic symbols. When encouraged, they can verbalize and explain the steps they perform, thereby demonstrating awareness of the procedures with symbols according to fixed rules. It is well known that…