**List of Contents**

**Summer 2015**

**Constructing Own-Featured Mathematics Instruction Theories**

*Ringbao Tu*

**Case Study on Statistics Lessons by High School Mathematics Teachers**

*Xie Yang Chun*

**Mathematical Education in Terms of Innovative Development**

*A. Pardala, R.A. Uteeva, N. K. Ashirbayev*

**Population and Exponential Growth Investigations: A Mathematical Lens**

*Brian Evans*

**Using Calculators in Teaching Calculus**

*Pavel Satyanov*

**An Observation Protocol Measuring Secondary Teachers’ Implementation of Dynamic Geometry Approach**

*M. Alejandra Sorto, Zhonghong Jiang, Alexander White, Sharon Strickland*

**Editorial: Concerns from Around the World**

We start this late summer issue with the two PPP from China by Professor Tu from Nanjing Normal University, till recently, the head of the National Chinese Mathematics. Association and by the teacher of mathematics Xie Yang Chun from… The presentations are to a certain degree complementary; While Tu introduces us to the basis of Chinese didactic thinking through the theoretical diagram: *classroom observation→case study→analytical framework→instruction design theories,* Chun presents a concrete case study along that theoretical approach. It is the study of the teacher of statistics in Gansu (?) who finds that the high school curriculum published by the Ministry of Education doesn’t formulate core concepts of statistics and therefore decides to find it by herself through the study of research literature and classroom observations of two teachers. Chun formulates her own theory on the “data analysis” as the core concept of high school statistics. At the same time Tu proceeds to state the core concept of the scientific development view on education to be the essential question: What kind of people we are going to educate? Tu’s presentation is the fascinating answer to that question.

As if the continuation of the theme Pardala, Uteeva and Ashirbayev representing the didactic thought of Poland, Russia and Kazachstan presents a concerned view upon the innovative development of teaching and quality of high school student preparation, motivation to study mathematics and mathematics oriented disciplines. Their suggestions for improvement of the situation are interesting. In particular, in Poland they suggest the need for collaboration between teachers and academicians as well as strong supports for in-service teachers in terms of increasing the quality of their work. In other words they see the need for “substantive” teaching-research. “Substantive” is the term formulated by Stenhouse in the seventies as the research whose main objective is to benefit others rather than a research community itself. Here the substantive task of teaching-research is to improve student learning in the classroom. Their discussion of mathematics education in Russia and Kazachstan is equally interesting. What is however surprising is that, in general comparisons are still made with respect to the West European math ed, whereas the best PISA, for example, results are obtained by Asiatic countries rather than West European.

Evans contribution introduces us to the issue of exponential growth of population both from real world point of view as well through the Bacteria thought experiment of Bartlet.

Satyanov paper makes the pitch for the important role of calculators in the mathematics classrooms – a familiar issue. Whereas one can wonder whether indeed “it is impossible to stop technological progress”, the examples of the power of calculators are very impressive.

The contribution of Sorto et al continues the concern for the proper use of technology in the classroom this time from the point of the implementation fidelity of Dynamic Geometry (DG) technology. It is a very precise, one could say, an elegant work with the NSF-supported assessment instrument. To continue being contrary in this issue, or playing “the devils advocate” role one could say it’s important that just as the precise instrument measuring fidelity of the teaching approach has been needed so that one doesn’t have to be “relying solely on teachers’ self-report when capturing implementation fidelity” one probably should not also rely solely on the instrument itself. What is important in this work for teaching-research is that the instrument ultimately measures teacher interest, capability and possibility within the curriculum to work with Discovery (or inquiry) method of teaching. The Discovery method plays a very important role for TR/NYCity Model which was developed in the community colleges of the Bronx in that it allows to interact and to investigate authentic student mathematical thinking. Maybe the interest of teachers in the implementation fidelity would increase if together with the didactic use of DG, they would be exposed to the investigations of student geometrical thinking with its help.