Using the Knowledge Quartet to quantify mathematical knowledge in teaching: the development of a protocol for Initial Teacher Education

This study examined trainee teachers' mathematical knowledge in teaching (MKiT) over their final year in a US Initial Teacher Education (ITE) programme. This paper reports on an exploratory methodological approach taken to use the Knowledge Quartet to quantify MKiT through the development of a new protocol to code trainees' teaching of mathematics lessons. This approach extends Rowland's et al. work on the Knowledge Quartet (KQ). Justification for using the KQ to quantify MKiT, and the potential benefits such an attempt might provide those involved with ITE, are discussed. It is suggested that quantified MKiT data based on the Knowledge Quartet can be used to consider MKiT development in novice teachers in order to inform ITE programmes and form new theoretical loops between theory and practice in teacher education.     

Partial sum process to check regression models with multiple correlated response: With an application for testing a change-point in profile data

We consider regression models with multiple correlated responses for each design point. Under the null hypothesis, a linear regression is assumed. For the least-squares residuals of this linear regression, we establish the limit of the partial sums. This limit is a projection on a certain subspace of the reproducing Kernel Hilbert space of a multivariate Brownian motion. Based on this limit, we propose a significance test of Kolmogorov-Smirnov type to test the null hypothesis and show that this result can be used to study a change-point problem in the case of linear profile data (panel data). We compare our proposed method, which does not rely on any distributional assumptions, with the likelihood ratio test in a simulation study.