Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

Snarks and Flow-Critical Graphs

It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutteʼs Flow Conjectures.     

Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph

We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus–Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.