In this study subjects are presented with sequences of heads and tails, derived from flipping a fair coin, and asked to consider their chances of occurrence. In this new iteration of the comparative likelihood task, the ratio of heads to tails in all of the sequences is maintained. In order to help situate participants’ responses within conventional probability, this article employs unconventional set descriptions of the sample space organized according to: switches, longest run, and switches and longest run, which are all based upon subjects’ verbal descriptions of the sample space. Results show that normatively incorrect responses to the task are not devoid of correct probabilistic reasoning. The notion of alternative set descriptions is further developed, and the article contends that sample space partitions can act as an investigative lens for research on the comparative likelihood task, and probability education in general.

This article deals with the interpretation of motion Cartesian graphs by Grade 8 students. Drawing on a sociocultural theoretical framework, it pays attention to the discursive and semiotic process through which the students attempt to make sense of graphs. The students’ interpretative processes are investigated through the theoretical construct of knowledge objectification and the configuration of mathematical signs, gestures, and words they resort to in order to achieve higher levels of conceptualization. Fine-grained video and discourse analyses offer an overview of the manner in which the students’ interpretations evolve into more condensed versions through the effect of what is called in the article “semiotic contractions” and “iconic orchestrations.”

In this article, we evaluate the true proportion of mathematics educators and teachers at under/post graduate levels in Karachi, Pakistan in making math courses lively to students. We use a random sample of 75 students of engineering and commerce studying in three different universities namely University of Karachi, Usman Institute of Technology (UIT) and Karachi Institute of Economics & Technology (PAF-KIET). A 95% confidence interval based on sample results reveals that the said proportion of math educators is in between 63 and 83%. Furthermore, we investigate with the help of students’ responses how mathematics teachers at under/post graduate levels make their courses interesting-by showing their dedication in their subject, by giving logical reasoning and concrete examples or by making complex mathematical methods accessible to students giving them know-how of mathematical softwares. We find that the second technique is the most dominant and has a very strong impact (positive linear relationship) in achieving the said goal of a math-teacher. The linear correlation coefficient between students’ opinion that math-teachers make their courses interesting and achieving this goal by giving logical reasoning and concrete examples is 0.989. Whereas the technique of using math softwares in attempt to make a math course lively has also a very strong but a cubic relationship and its multiple correlation coefficient is 0.984. Therefore, using technology in math classroom is also helpful in making math learning and teaching interesting but under some conditions that become apparent from our study made on the real data hence obtained.

In this article, we aim to provide a glimpse of what is counted as good mathematics instruction from Taiwanese perspectives and of various approaches developed and used for achieving high-quality mathematics instruction. The characteristics of good mathematics instruction from Taiwanese perspectives were first collected and discussed from three types of information sources. Although the number of characteristics of good mathematics instruction may vary from one source to another, they can be generally organized in three phases including lesson design before instruction, classroom instruction during the lesson and activities after lesson. In addition to the general overview of mathematics classroom instruction valued in Taiwan, we also analyzed 92 lessons from six experienced teachers whose instructional practices were generally valued in local schools and counties. We identified and discussed the characteristics of their instructional practices in three themes: features of problems and their uses in classroom instruction, aspects of problem–solution discussion and reporting, and the discussion of solution methods. To identify and promote high-quality mathematics instruction, various approaches have been developed and used in Taiwan including the development and use of new textbooks and teachers’ guides, teaching contests, master teacher training program, and teacher professional development programs.

In this study, selected Chinese, Japanese and US mathematics textbooks were examined in terms of their ways of conceptualizing and organizing content for the teaching and learning of fraction division. Three Chinese mathematics textbook series, three Japanese textbook series, and four US textbook series were selected and examined to locate the content instruction of fraction division. Textbook organization of fraction division and other content topics were described. Further analyses were then conducted to specify how the content topic of fraction division was conceptualized and introduced. Specific attention was also given to the textbooks’ uses of content constructs including examples, representations, and exercise problems in order to show their approaches for the teaching and learning of fraction division. The results provide a glimpse of the metaphors of mathematics teaching and learning that have been employed in Chinese, Japanese, and US textbooks. In particular, the results from the textbook analyses demonstrate how conceptual underpinnings were developed while targeting procedures and operations. Implications of the study are then discussed.

• The use of language in understanding subject matter, by Lennart Svensson, Elsie Anderberg, Christer Alvegård and Thorsten Johansson
• Online but off-topic: negotiating common ground in small learning groups, by Trena M. Paulus
• The building of pre-service primary teachers’ knowledge of mathematics teaching: interaction and online video case studies, by Salvador Llinares and Julia Valls
• The learners’ experience of variation: following students’ threads of learning physics in computer simulation sessions, by Åke Ingerman, Cedric Linder and Delia Marshall
• The development of science activities via on-line peer assessment: the role of scientific epistemological views, by Chin-Chung Tsai and Jyh-Chong Liang

• Acquisition and use of shortcut strategies by traditionally schooled children, by Joke Torbeyns, Bert De Smedt, Pol Ghesquière and Lieven Verschaffel
• From arithmetical thought to algebraic thought: The role of the “variable”, by Elsa Malisani and Filippo Spagnolo
• The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish, by Máire Ní Ríordáin and John O’Donoghue
• Teachers’ emergent goals in spreadsheet-based lessons: analyzing the complexity of technology integration, by Jean-Baptiste Lagrange and Emel Ozdemir Erdogan
• Book review: mathematics classrooms in twelve countries, Clarke, D., Keitel, C., & Shimizu, Y. (Eds.). (2006). Mathematics classrooms in twelve countries: The insider’s perspective. Rotterdam, The Netherlands: Sense Publishers.