One of the most significant steps in learning algebra is understanding the change in the role of letters in mathematical expressions from unknowns to variables. We describe the historical development of this change in usage, starting with the ancient use of mathematical unknowns, detailing several important changes in practice that allowed for the…

Graphical tasks have become a prominent aspect of mathematics assessment. From a conceptual stance, the purpose of this study was to better understand the composition of graphical tasks commonly used to assess students' mathematics understandings. Through an iterative design, the investigation described the sense making of 11-12-year-olds as they…

Undergraduate physical chemistry courses require students to be proficient in calculus in order to develop an understanding of thermodynamics concepts. Here we present the findings of a study that examines student understanding of mathematical expressions, including partial derivative expressions, in two undergraduate physical chemistry courses.…

Despite an emphasis on manipulative algebraic techniques in secondary school algebra, many tertiary mathematics students have mastered these skills without conceptual understanding. A significant number of students with high tertiary entrance ranks enrolled in first semester university mathematics were found to have misconceptions relating to…

The purpose of this study is to examine the nature of what students notice about symbols and use as they solve unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions. The researcher conducted a study of students at three levels of algebra exposure: (a) students enrolled in a high school pre-calculus…

In this article, the authors begin by considering symbolic literacy in mathematics. Next, they examine the origins of misuse of the equals sign by primary and junior secondary students, where "=" has taken on an operational meaning. They explain that in algebra, students need both the operational and relational meanings of the equals…

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation…

There has been continuing pressure to incorporate algebraic thinking into K-12 education. Elementary teachers will be expected to incorporate and successfully execute algebra related lessons. This dissertation study examines pre-service elementary teachers' conceptions and content knowledge regarding literal symbols and their usage. A written…

Symbolic power is discussed with reference to mathematics and formal languages. Two distinctions are crucial for establishing mechanical and formal perspectives: one between appearance and reality, and one between sense and reference. These distinctions include a nomination of what to consider primary and secondary. They establish the grammatical…

The National Council of Teachers of Mathematics [NCTM] noted that middle and high school students are expected to be able to both explain inequalities by using mathematical symbols and understand meanings by interpreting the solutions of inequalities. Unfortunately, research has revealed that not only do middle and high school students hold…

It is not surprising, as research has shown, that fractions are one of the most difficult of the elementary school math topics to teach and learn in ways that are meaningful. The authors reference a work by James Hiebert, "Mathematical, Cognitive, and Instructional Analyses of Decimal Fractions" (1992), that mathematical concepts should…

Numerical understanding and arithmetic skills are easier to acquire for whole numbers than fractions. The "integrated theory of numerical development" posits that, in addition to these differences, whole numbers and fractions also have important commonalities. In both, students need to learn how to interpret number symbols in terms of…

Difficulties in general education mathematics courses may be attributed to many factors, primarily low proficiency in symbol manipulation, a perception that mathematics is an area which eludes mastery, a lack of engagement and effective practice. Educational technology can be a powerful aid in overcoming these factors. This work describes the…

This article defines the concepts related to symbolic and sign representations, cognition and learning in the early years. The first study experiment of teaching 33 preschool children (19 boys and 14 girls; M = 68, 5 months) the notion of rainbow phenomenon proved the equal effectiveness of the use of both sign and symbolic tools. The second study…

Are numerals estranged from a sense of the actual quantities they represent? We demonstrate that, irrespective of numerical size or distance, direct comparison of the relative quantities represented by symbolic and nonsymbolic formats leads to performance markedly worse than when comparing 2 nonsymbolic quantities (Experiment 1). Experiment 2…