In this paper we interpret pervasive difficulties with second degree inequalities as a macro-phenomenon emerging from Large Scale Assessment. In line with the theory of reification, we link common students' difficulties to approaches they use that undermine the intertwining of processes and the ensuing emergent mathematical objects. To frame the…

The transition from arithmetic to algebra is a well-known difficulty in school mathematics. In order to succeed, learners require inter alia a better understanding of algebraic symbols, equality, equations and working with negatives/subtraction. This article reports on a response pattern analysis (RPA) of learners' responses to six test items…

In the subject matter of functional relationships, a student's ability to translate from one form of representation to another is seen as a central competence. In the course of the HEUREKO project (heuristic work with representations of functional relationships and the diagnosis of mathematical competencies of students), a theoretical competence…

The objective of the present investigation was to compare the adequacy of two cognitive models for predicting examinee performance on a sample of algebra I and II items from the March 2005 administration of the SAT[TM]. The two models included one generated from verbal reports provided by 21 examinees as they solved the SAT[TM] items, and the…

Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…

One possible approach students can cope with abstract algebra concepts is reducing abstraction. This notion occurs when learners are unable to adopt mental strategies as they deal with abstraction level of a given task. To make these concepts mentally accessible for themselves, learners unconsciously reduce the level of the abstraction of the…

To probe beyond declarative knowledge about real numbers in secondary school, the authors interviewed students in Grades 9, 10 and 12. The main question seems to be whether the length of a decimal expansion is "indefinite" or "infinite." It blurs the mental representation of rational numbers as well. (Contains 3 footnotes.) [For complete…