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This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions.

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A soluble phospholipase C (PLC) from boar sperm generates InsP(3) and hence causes Ca(2+) release when added to sea urchin egg homogenate. This PLC activity is associated with the ability of sperm extracts to cause Ca(2+) oscillations in mammalian eggs following fractionation. A sperm PLC may, therefore, be responsible for causing the observed Ca(2+) oscillations at fertilization. In the present study we have further characterized this boar sperm PLC activity using sea urchin egg homogenate. Consistent with a sperm PLC acting on egg PtdIns(4,5)P(2), the ability of sperm extracts to release Ca(2+) was blocked by preincubation with the PLC inhibitor U73122 or by the addition of neomycin to the homogenate. The Ca(2+)-releasing activity was also detectable in sperm from other species and in whole testis extracts. However, activity was not observed in extracts from other tissues. Moreover recombinant PLCbeta1, -gamma1, -gamma2, -delta1, all of which had higher specific activities than boar sperm extracts, were not able to release Ca(2+) in the sea urchin egg homogenate. In addition these PLCs were not able to cause Ca(2+) oscillations following microinjection into mouse eggs. These results imply that the sperm PLC possesses distinct properties that allow it to hydrolyse PtdIns(4,5)P(2) in eggs.

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While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students' understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Prestructural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.

Given the important role played by students' spatial reasoning skills, in this paper we analyse how students use these skills to solve problems involving 2D representations of 3D geometrical shapes. Using data from in total 1357 grades 4 to 9 students, we examine how they visualise shapes in the given diagrams and make use of properties of shapes to reason. We found that using either spatial visualisation or property-based spatial analytic reasoning is not enough for the problems that required more than one step of reasoning, but also that these two skills have to be harmonised by domainspecific knowledge in order to overcome the perceptual appearance (or "look") of the given diagram. We argue that more opportunities might be given to both primary and secondary school students in which they can exercise not only their spatial reasoning skills but also consolidate and use their existing domain-specific knowledge of geometry for productive reasoning in geometry.