Kiril Bankov

The Intended, Implemented, and Achieved Curriculum of Mathematics Teacher Education in the United States

The first part of this chapter describes the development of a systematic model for the comparative study of the curriculum of teacher education programs used in the Teacher Education and Development Study in Mathematics (TEDS-M). We use the United States as a case study to describe in depth the intended, implemented, and achieved curriculum of teacher education. We present in detail the results of the curriculum analysis for programs preparing future primary and secondary teachers (or the intended curriculum). We contrast these results with the number of clock hours dedicated to different areas of teacher education, and with future teachers’ perception of their opportunities to learn (as indicators of the implemented curriculum), and with future teachers’ knowledge at graduation (as indicators of the achieved curriculum).

Arrangements and Transformations of Numbers on a Circle: An Essay Inspired by Problems of Mathematics Competitions

There are contest problems dealing with the following situation: several numbers are arranged on a circle and a certain admissible operation can be consecutively done finite number of times; the task is to find conditions under which a specific final arrangement of the numbers can be obtained. The variety of these problems is determined by different initial and final arrangements of the numbers and by the admissible operations with them. The change of some of these elements often leads to interesting generalizations. This chapter discusses several such examples. It also presents some other contest problems dealing with arrangements of numbers on a circle. Didactical approaches to teaching how to solve such problem are also considered.

Bulgaria: Piloting Statistical Models for Estimation of Schools’ Value-Added Using the Results from the National Assessments

The objective of this report is to document the process and summarize the results from the second phase of the pilot implementation of statistical models for measuring the value-added of Bulgarian schools through analysis of the national student’ assessments results. This report presents the technical aspects of the pilot and the key outcomes in terms of value-added measure for each of the schools included in the analysis. It is intended to ensure the institutional memory for the pilots and to stimulate the discussions at technical level among experts and decision makers at the Ministry of Education and Science (MES), focusing on the relevance and applicability of the piloted statistical models, in the context of Bulgaria student assessment framework, and given the available data. The report documents all the data processing, adjustments, and procedures run as part of the school value added modeling.

PRINCIPLE OF THE EXTREME ELEMENT

Many professional mathematicians wish to contribute to making mathematics more attractive to gifted young people. One way to pursue this goal is to create of problems that require a great deal of common sense, imagination, and, very often, a specific problem solving strategy. This article introduces such a strategy, the principle of the extreme element. Although the name is not widely used, the principle may help you to solve some non-standard mathematical problems. The material is based on my personal experience gained in extracurricular work with mathematically gifted students, and in my participation in mathematics competitions at different levels as a contestant and organiser.