Developmental pathway for individualized learning of algebra

Abstract

While much research has been conducted into student difficulty in understanding and processing algebra, learning difficulties in these areas persist. Also, problems continue to be encountered in supporting students to progress from concrete to abstract reasoning, and the related concern encountered by learners when engaging in higher-order thinking in algebra. As background to the data collection, four cognitive gaps in student thinking were identified from the research literature. These cognitive gaps existed between: arithmetic and algebra; procedural and conceptual knowledge; processing skills; and concrete and abstract ideas. Of significance to this study (and particularly to the latter two cognitive gaps) was research focusing on the relationship between the capability of students to solve higher-order problems and those students’ capability to use the processing skills of reversibility, flexibility, and generalization (Krutetskii, 1976). The existence and elaboration of cognitive gaps was further explored through an analysis of New South Wales Higher School Certificate Examinations (highstakes end-of-secondary-schooling examinations) in Mathematics over a period of ten years. NSW offers three hierarchical calculus-based courses and two lessintensive algebra-based courses in senior secondary education. The senior markers' comments for all courses were consistent across different years and showed that these four cognitive gaps were strongly evident within student responses to examination questions, including from those students who undertook the more demanding calculus-based courses. Confirmation of these cognitive gaps laid the foundation for the research. The aims of the study, built on the research literature and the analysis of examiner comments, were to: (i) explore the nature of the cognitive impediments that might be responsible for student difficulty in understanding and processing a specific algebraic topic; and (ii) discover an approach within the teaching of algebra that might overcome impediments originating from within the subject algebra and from within student cognition. The design of the research evolved over the life of the study with the original focus continually being refined. These changes can be described under two phases, with the first phase providing the background, and a tighter research focus, for the second phase. The first phase involved building a developmental series of four question parts for each of ten domains of algebra that was administered as a student writtenresponse assessment. Analysis of these responses indicated a range of errors made, and difficulties experienced, by secondary school students that were then summarized as five broad impediments to higher-order thinking in algebra. The second phase consisted of cognitive conversations between the researcher and a sample of students drawn from across three ability levels. The purpose was to investigate the impact of engaging with the processing skills reversibility, flexibility, and generalization on developing higher-order thinking. A lack of student engagement with the three processing skills prompted adjustments in the cognitive conversation to focus more specifically on: what was preventing students from understanding and processing higher-order question parts; and the role played by reversibility, flexibility, and generalization in higher-order thinking. These adjustments to the cognitive conversations revealed that most impediments were associated with three areas of understanding algebra: procedural and conceptual knowledge; image-supported and image-independent reasoning; and inter-question-part functioning. Regarding the first aim of the study, the main factors found to be preventing understanding and processing higher-order question parts were reliance on procedural approaches with no underlying conceptual understanding, and use of either visual or non-visual logic separately, rather than in supportive combination. Although reversibility, flexibility, and generalization processing skills were not found to be as useful as expected in supporting achievement in higher-order question parts, they were found to be an important indicator of the level of a student's understanding. Regarding the second aim, the main findings from the study enabled the creation of a learning approach or path that provides a basis from which to help overcome identified impediments to cognitive development students are encountering. Progression along this learning path provided a structure for a model of algebraic problem solving. The application of this model supports students to develop higher-order thinking through: a mechanism for overcoming the five broad impediments and specific impediments associated with the contextual and procedural demands of questions; ways of transforming an impediment into a catalyst for new actions; and developing the three areas of understanding algebra to develop enhanced perspectives suitable for solving higher-order problems. Suggested implications of the study findings impact on both assessment practices, and teaching and learning practices. The challenge for assessment practices is to use a student's understanding of the reason an answer is correct as a more informative indicator of students' understanding than simply relying on the fact that the student has written down a correct answer. The challenge for teaching and learning practices is for teachers to develop approaches on how new problems can be presented to students by building on similar lower-order problems previously solved and understood. This approach enables students to relate to, and process, new contexts and new concepts more independently.