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There is a growing consensus that algebra is an important aspect of mathematics teaching and learning and several abilities are required in order students to have successful performance in algebra. The present study uses insights from the domain of psychology to enrich what is currently known in the domain of mathematics education about the relationship of algebraic thinking with abilities involved in fundamental cognitive processes. In total, 190 students between the ages of 13–17 years old were tested through two tests. The first test addressed four types of cognitive systems which are responsible for the representation and processing of different types of relations in the environment: the spatial-imaginal, the causal-experimental, the qualitative-analytic and the verbal-propositional. The second test addressed algebraic thinking. The results support the key role of the four types of cognitive processes in students’ algebraic thinking. The results also suggest that abilities involved in the four types of cognitive processes predict algebraic thinking abilities, irrespective of the age of the students. [ABSTRACT FROM PUBLISHER]

This paper presents the partial results of a research on the professional knowledge of mathematics olympiad evaluators teachers put into action when analyzing errors relating to algebra. By means of two tests on errors interpretation and by using analysis inspired by the grounded theory, we obtained dimensions for the characterization of knowledge in understanding about errors. In the results section, we describe a teacher's knowledge related to practice and knowledge about students' learning. [ABSTRACT FROM AUTHOR]

Adaptive learning technologies are emerging in educational settings as a means to customize instruction to learners' background, experiences, and prior knowledge. Here, a technology-based personalization intervention within an intelligent tutoring system (ITS) for secondary mathematics was used to adapt instruction to students' personal interests. We conducted a learning experiment where 145 ninth-grade Algebra I students were randomly assigned to 2 conditions in the Cognitive Tutor Algebra ITS. For 1 instructional unit, half of the students received normal algebra story problems, and half received matched problems personalized to their out-of-school interests in areas such as sports, music, and movies. Results showed that students in the personalization condition solved problems faster and more accurately within the modified unit. The impact of personalization was most pronounced for 1 skill in particular--writing symbolic equations from story scenarios--and for 1 group of students in particular--students who were struggling to learn within the tutoring environment. Once the treatment had been removed, students who had received personalization continued to write symbolic equations for normal story problems with increasingly complex structures more accurately and with greater efficiency. Thus, we provide evidence that interest-based interventions can promote robust learning outcomes--such as transfer and accelerated future learning--in secondary mathematics. These interest-based connections may allow for abstract ideas to become perceptually grounded in students' experiences such that they become easier to grasp. Adaptive learning technologies that utilize interest may be a powerful way to support learners in gaining fluency with abstract representational systems. [ABSTRACT FROM AUTHOR]

Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students' errors, less is known about what students do understand to be general and convincing. This study examines the ways in which seven middle school students generalized and justified while exploring linear functions. Students' generalizations and proof schemes were identified and categorized in order to establish connections between types of generalizations and types of justifications. These connections led to the identification of four mechanisms for change that supported students' engagement in increasingly sophisticated forms of algebraic reasoning: (a) iterative action/reflection cycles, (b) mathematical focus, (c) generalizations that promote deductive reasoning, and (d) influence of deductive reasoning on generalizing. [ABSTRACT FROM AUTHOR]