Making Upper Division Mathematics Courses Relevant for Pre-Service Teachers

This article addresses the disconnect that in-service and pre-service secondary school teachers feel between the material presented in upper division mathematics courses and high school classroom practice. Two examples are given from an abstract algebra course in which this problem is addressed

Connecting High School Mathematics and Abstract Algebra

This thesis focuses on the connections between the content covered in college level abstract algebra and high school mathematics courses. The purpose of studying and understanding these connections is in hopes to improve the way concepts are presented at the high school level to give students a deeper understanding of the mathematics. High school mathematics teachers are required to take specific college courses such as abstract algebra before they can teach high school mathematics. However, many teachers do not believe they can use the complex ideas learned in college to teach their students. On the surface, the content from courses like abstract algebra may seem unrelated, but recognizing important links and altering instruction could make college level mathematical ideas seem accessible to the high school students. We will not only study the connections in content, but also examine different ways to teach the content and how this can be beneficial to students

Third-graders' problem solving performance and teachers' actions

Peer reviewe

Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe

Routines and Applications of Symbolic Algebra Software

Computing has become an essential resource in modern research and has found application across a wide range of scientific disciplines. Developments in symbolic algebra tools have been particularly valuable in physics where calculations in fields such as general relativity, quantum field theory and physics beyond the standard model are becoming increasing complex and unpractical to work with by hand. The computer algebra system Cadabra is a tensor-first approach to symbolic algebra based on the programming language Python which has been used extensively in research in these fields while also having a shallow learning curve making it an excellent way to introduce students to methods in computer algebra. The work in this thesis has been concentrated on developing Cadabra, which has involved looking at two different elements which make up a computer algebra program. Firstly, the implementation of algebraic routines is discussed. This has primarily been focused on the introduction of an algorithm for detecting the equivalence of tensorial expressions related by index permutation symmetries. The method employed differs considerably from traditional canonicalisation routines which are commonly used for this purpose by using Young projection operators to make such symmetries manifest. The other element of writing a computer algebra program which is covered is the infrastruc- ture and environment. The importance of this aspect of software design is often overlooked by funding committees and academic software users resulting in an anti-pattern of code not being shared and contributed to in the way in which research itself is published and promulgated. The focus in this area has been on implementing a packaging system for Cadabra which allows the writing of generic libraries which can be shared by the community, and interfacing with other scientific computing packages to increase the capabilities of Cadabra