The 'algebra as object' analogy: a view from school

Treating algebraic symbols as objects (eg. “‘a’ means ‘apple’”) is a means of introducing elementary simplification of algebra, but causes problems further on. This current school-based research included an examination of texts still in use in the mathematics department, and interviews with mathematics teachers, year 7 pupils and then year 10 pupils asking them how they would explain, “3a + 2a = 5a” to year 7 pupils. Results included the notion that the ‘algebra as object’ analogy can be found in textbooks in current usage, including those recently published. Teachers knew that they were not ‘supposed’ to use the analogy but not always clear why, nevertheless stating methods of teaching consistent with an‘algebra as object’ approach. Year 7 pupils did not explicitly refer to ‘algebra as object’, although some of their responses could be so interpreted. In the main, year 10 pupils used ‘algebra as object’ to explain simplification of algebra, with some complicated attempts to get round the limitations. Further research would look to establish whether the appearance of ‘algebra as object’ in pupils’ thinking between year 7 and 10 is consistent and, if so, where it arises. Implications also are for on-going teacher training with alternatives to introducing such simplification.

Comment on ``Piecewise potential vorticity inversion: Elementary tests''

Egger (2008) constructs some idealised experiments to test the usefulness of piecewise potential vorticity inversion (PPVI) in the diagnosis of Rossby wave dynamics and baroclinic development. He concludes that, ``PPVI does not help us to understand the dynamics of linear Rossby waves. It provides local tendencies of the streamfunction which are unrelated to the true ones. The same way, the motion of baroclinic waves in shear flow cannot be understood by using PPVI. Moreover, the effect of boundary temperatures as determined by PPVI is unrelated to the flow evolution.'' He goes further in arguing that we should not consider velocities as ``induced'' by PV anomalies defined by carving up the global domain. However, these conclusions partly reflect the limitations of his idealised experiments and the manner in which the PV components were partitioned from one another.

Monte Carlo numerical treatment of large linear algebra problems

In this paper we deal with performance analysis of Monte Carlo algorithm for large linear algebra problems. We consider applicability and efficiency of the Markov chain Monte Carlo for large problems, i.e., problems involving matrices with a number of non-zero elements ranging between one million and one billion. We are concentrating on analysis of the almost Optimal Monte Carlo (MAO) algorithm for evaluating bilinear forms of matrix powers since they form the so-called Krylov subspaces. Results are presented comparing the performance of the Robust and Non-robust Monte Carlo algorithms. The algorithms are tested on large dense matrices as well as on large unstructured sparse matrices.

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.

Measuring students' emotions in the early years: the Achievement Emotions Questionnaire-Elementary School (AEQ-ES)

This article reports about the development and validation of a measurement instrument assessing elementary school students' achievement emotions (Achievement Emotions Questionnaire-Elementary School, AEQ-ES). Specifically, the instrument assesses students' enjoyment, anxiety, and boredom pertaining to three types of academic settings (i.e., attending class, doing homework, and taking tests and exams). Scale construction was based on Pekrun's (2006) control-value theory of achievement emotions. The instrument was tested using samples from German and American elementary school classrooms. The results of Study 1 (German sample) corroborate the reliability and structural validity of the new emotion measure. Moreover, they show that students' achievement emotions were linked with their control and value appraisals as well as their academic performance, thus supporting the external validity of the measure as well as propositions of Pekrun's (2006) control-value theory of achievement emotions. Study 2 (American sample) corroborated the cross-cultural equivalence of the measure and the generalizability of findings across the German and American samples. Implications for research on achievement emotions and educational practice are discussed. (PsycINFO Database Record (c) 2013 APA, all rights reserved)(journal abstract)

Transreal limits and elementary functions

We extend all elementary functions from the real to the transreal domain so that they are defined on division by zero. Our method applies to a much wider class of functions so may be of general interest.