Deficiencies in beginning undergraduate students' basic mathematical skills has been an issue of concern in higher education, particularly in the past 15 years. This issue has been tracked and analysed in a number of universities in Ireland and internationally through student scores recorded in mathematics diagnostic tests. Students beginning their science-based and technology-based undergraduate courses in the University of Limerick have had their basic mathematics skills tested without any prior warning through a 40 question diagnostic test during their initial service mathematics lecture since 1998. Data gathered through this diagnostic test have been recorded in a database kept at the university and explored to track trends in mathematical competency of these beginning undergraduates. This paper details findings surrounding an analysis of the database between 2003 and 2013, outlining changes in mathematical competencies of these beginning undergraduates in an attempt to determine reasons for such changes. The analysis found that the proportion of students tested through this diagnostic test that are predicted to be at risk of failing their service mathematics end-of-semester examinations has increased significantly between 2003 and 2013. Furthermore, when students' performance in secondary level mathematics was controlled, it was determined that the performance of beginning undergraduates in 2013 was statistically significantly below that of the performance of the beginning undergraduates recorded 10 years previously.
Algebra has long been identified as an area of difficulty in the teaching and learning of mathematics. Evidence of this difficulty can be found in Irish secondary level classrooms. Chief Examiner Reports have consistently identified algebra as an area of student weakness in State examinations. In light of poor student performance, and as part of a nationwide reform of secondary mathematics curricula, a functions based approach to teaching algebra has been adopted in Irish schools. It was introduced in September 2011 in place of the transformational (rule and procedure) based approach which was previously used. Through comparing the diagnostic test scores of incoming students in an Irish university in the years before and after the reform, this study finds that the reformed approach has coincided with a decline in students' technical algebraic skills. However interviews with practicing mathematics teachers reveal that this decline is not a direct result of the functions based approach, but rather of a mixture of approaches being implemented in classrooms. Such divergence of approaches can be linked to the common mismatch between the intended curriculum prescribed by policy makers and the implemented curriculum that is actually carried out by teachers in their classrooms.
This article presents a theoretical model for integrating mathematics and science in the secondary classroom. This model, Authentic Integration of Mathematics and Science (AIMS), promotes engagement with rich tasks which combine topics from mathematics and science to enable enhanced learning through structured inquiry, dialogue, and application of knowledge and skills from both subjects to relatable tasks. It is argued that this model will provide opportunities for students to retrieve previously learned material and explore key concepts from both disciplines in tandem, thereby strengthening retention and understanding. Application of this model should also support the enhancement of students' problem-solving skills and the facilitation of meaningful applications of mathematics to other disciplines in a sustainable manner. Attempts to integrate mathematics and science in the classroom are widely recommended but often encounter barriers such as deficiencies in teacher knowledge of their non-specialist subject, the inflexible nature of school timetables, and a dearth of instructional materials, amongst other issues. Lesson study is proposed as an effective means for operationalising the AIMS model and providing a framework which accounts for these barriers and allows for consistent implementation in tandem with single-subject instruction.
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