Developing geometrical reasoning in the secondary school: outcomes of trialling teaching activities in classrooms, a report to the QCA

This report presents the findings of the Southampton/Hampshire Group of mathematicians and mathematics educators sponsored by the Qualifications and Curriculum Authority (QCA) to develop and trial some teaching/learning materials for use in schools that focus on the development of geometrical reasoning at the secondary school level. The project ran from October 2002 to November 2003. An interim report was presented to the QCA in March 2003. 1. The Southampton/Hampshire Group consisted of five University mathematicians and mathematics educators, a local authority inspector, and five secondary school teachers of mathematics. The remit of the group was to develop and report on teaching ideas that focus on the development of geometrical reasoning at the secondary school level. 2. In reviewing the existing geometry curriculum, the group endorsed the RS/ JMC working group conclusion (RS/ JMC geometry report, 2001) that the current mathematics curriculum for England contains sufficient scope for the development of geometrical reasoning, but that it would benefit from some clarification in respect of this aspect of geometry education. Such clarification would be especially helpful in resolving the very odd separation, in the programme of study for mathematics, of ‘geometrical reasoning’ from ‘transformations and co-ordinates’, as if transformations, for example, cannot be used in geometrical reasoning. 3. The group formulated a rationale for designing and developing suitable teaching materials that support the teaching and learning of geometrical reasoning. The group suggests the following as guiding principles: • Geometrical situations selected for use in the classroom should, as far as possible, be chosen to be useful, interesting and/or surprising to pupils; • Activities should expect pupils to explain, justify or reason and provide opportunities for pupils to be critical of their own, and their peers’, explanations; • Activities should provide opportunities for pupils to develop problem solving skills and to engage in problem posing; • The forms of reasoning expected should be examples of local deduction, where pupils can utilise any geometrical properties that they know to deduce or explain other facts or results. • To build on pupils’ prior experience, activities should involve the properties of 2D and 3D shapes, aspects of position and direction, and the use of transformation-based arguments that are about the geometrical situation being studied (rather than being about transformations per se); • The generating of data or the use of measurements, while playing important parts in mathematics, and sometimes assisting with the building of conjectures, should not be an end point to pupils’ mathematical activity. Indeed, where sensible, in order to build geometric reasoning and discourage over-reliance on empirical verification, many classroom activities might use contexts where measurements or other forms of data are not generated. 4. In designing and trialling suitable classroom material, the group found that the issue of how much structure to provide in a task is an important factor in maximising the opportunity for geometrical reasoning to take place. The group also found that the role of the teacher is vital in helping pupils to progress beyond straightforward descriptions of geometrical observations to encompass the reasoning that justifies those observations. Teacher knowledge in the area of geometry is therefore important. 5. The group found that pupils benefit from working collaboratively in groups with the kind of discussion and argumentation that has to be used to articulate their geometrical reasoning. This form of organisation creates both the need and the forum for argumentation that can lead to mathematical explanation. Such development to mathematical explanation, and the forms for collaborative working that support it, do not, however, necessarily occur spontaneously. Such things need careful planning and teaching. 6. Whilst pupils can demonstrate their reasoning ability orally, either as part of group discussion or through presentation of group work to a class, the transition to individual recording of reasoned argument causes significant problems. Several methods have been used successfully in this project to support this transition, including 'fact cards' and 'writing frames', but more research is needed into ways of helping written communication of geometrical reasoning to develop. 7. It was found possible in this study to enable pupils from all ages and attainments within the lower secondary (Key Stage 3) curriculum to participate in mathematical reasoning, given appropriate tasks, teaching and classroom culture. Given the finding of the project that many pupils know more about geometrical reasoning than they can demonstrate in writing, the emphasis in assessment on individual written response does not capture the reasoning skills which pupils are able to develop and exercise. Sufficient time is needed for pupils to engage in reasoning through a variety of activities; skills of reasoning and communication are unlikely to be absorbed quickly by many students. 8. The study suggests that it is appropriate for all teachers to aim to develop the geometrical reasoning of all pupils, but equally that this is a non-trivial task. Obstacles that need to be overcome are likely to include uncertainty about the nature of mathematical reasoning and about what is expected to be taught in this area among many teachers, lack of exemplars of good practice (although we have tried to address this by lesson descriptions in this report), especially in using transformational arguments, lack of time and freedom in the curriculum to properly develop work in this area, an assessment system which does not recognise students’ oral powers of reasoning, and a lack of appreciation of the value of geometry as a vehicle for broadening the curriculum for high attainers, as well as developing reasoning and communication skills for all students. 9. Areas for further work include future work in the area of geometrical reasoning, include the need for longitudinal studies of how geometrical reasoning develops through time given a sustained programme of activities (in this project we were conscious that the timescale on which we were working only enabled us to present 'snapshots'), studies and evaluation of published materials on geometrical reasoning, a study of 'critical experiences' which influence the development of geometrical reasoning, an analysis of the characteristics of successful and unsuccessful tasks for geometrical reasoning, a study of the transition from verbal reasoning to written reasoning, how overall perceptions of geometrical figures ('gestalt') develops as a component of geometrical reasoning (including how to create the links which facilitate this), and the use of dynamic geometry software in any (or all) of the above.10. As this group was one of six which could form a model for part of the work of regional centres set up like the IREMs in France, it seems worth recording that the constitution of the group worked very well, especially after members had got to know each other by working in smaller groups on specific topics. The balance of differing expertise was right, and we all felt that we learned a great deal from other group members during the experience. Overall, being involved in this type of research and development project was a powerful form of professional development for all those concerned. In retrospect, the group could have benefited from some longer full-day meetings to jointly develop ideas and analyse the resulting classroom material and experience rather than the pattern of after-school meetings that did not always allow sufficient time to do full justice to the complexity of many of the issues the group was tackling

The experimental degradation of archaeological human bone by anaerobic bacteria and the implications for recovery of ancient DNA

DNA recovery from human bone has been key to the developing science and technology of ancient DNA studies. The recovery of macromolecules from bone however, does not correlate well with recognisable parameters of preservation and predicting DNA recovery rates from ancient bone can be very difficult. The extent of degradation of buried bones often depends on environmental taphonomy and can vary from virtually none to complete and rapid destruction. Although soil or related microbes are undoubtedly responsible for the majority of this structural degradation over time, exceptionally little is known of the mechanisms or specific bacteria involved. Fungi were previously thought to be responsible for destructive processes (tunnelling) within bone but over the last 30 years the role of bacteria has been increasingly recognised. Our aim was to develop a less complex in vitro model of the destructive effects of microbes on bone which might allow a better understanding of the recovery of mitochondrial or pathogen DNA over time

Feature Selection Using Regularization in Approximate Linear Programs for Markov Decision Processes

Approximate dynamic programming has been used successfully in a large variety of domains, but it relies on a small set of provided approximation features to calculate solutions reliably. Large and rich sets of features can cause existing algorithms to overfit because of a limited number of samples. We address this shortcoming using $L_1$ regularization in approximate linear programming. Because the proposed method can automatically select the appropriate richness of features, its performance does not degrade with an increasing number of features. These results rely on new and stronger sampling bounds for regularized approximate linear programs. We also propose a computationally efficient homotopy method. The empirical evaluation of the approach shows that the proposed method performs well on simple MDPs and standard benchmark problems.Comment: Technical report corresponding to the ICML2010 submission of the same nam

Work in progress: a novel method of creating an academic content repository

This paper outlines a project aimed at addressing the issue of the scalability of online academic support. This project is being run during the Autumn semester at the University of Southern Queensland (USQ) Toowoomba Australia, in conjunction with the Australian Digital Futures Institute. The study attached to the project will use Design-Based Research to evaluate the effectiveness of a simple, but innovative academic content and metadata creation tool referred to as Academic Assist. Academic Assist has been recently developed at USQ as a plug-in block for the moodle-based Learning Management System employed at USQ for its several hundred online subjects. The pilot project and associated study now extends over nine subjects, including three consecutive subjects in computer engineering; and covers faculties of Engineering, Education, Business, Science and Arts. Some preliminary results are presented here. Complete results of the study including acceptance surveys, expert reviews and usage statistics will be presented at FIE 2009

The Quest for Education: Racism, Paradox, and Interest Convergence in the Life of George Washington Carver

George Washington Carver is known primarily for his life and work at Tuskegee Institute in Alabama. This historical research provides insight prior to that time and into his journey from Missouri to several towns in Kansas, and then to Iowa. The intersection of race, education, and philanthropy combined to guide the culmination of his life’s work—in concert with interest convergence

LibQuest: A Web App for Exploring the IC Library

LibQuest is a web app that is designed to help students explore the physical space of the library and can be used in place of a library scavenger hunt or tour. Questions encourage students to interact with service points and physical collections, and are randomized to discourage copying. Text-input questions are included to increase difficulty, as are questions that can only be answered by visiting the building

Hollywood Free Paper, June 15 1971

https://digitalcommons.fuller.edu/hollywoodfreepaper/1043/thumbnail.jp

Comparing Probabilistic Models for Melodic Sequences

Modelling the real world complexity of music is a challenge for machine learning. We address the task of modeling melodic sequences from the same music genre. We perform a comparative analysis of two probabilistic models; a Dirichlet Variable Length Markov Model (Dirichlet-VMM) and a Time Convolutional Restricted Boltzmann Machine (TC-RBM). We show that the TC-RBM learns descriptive music features, such as underlying chords and typical melody transitions and dynamics. We assess the models for future prediction and compare their performance to a VMM, which is the current state of the art in melody generation. We show that both models perform significantly better than the VMM, with the Dirichlet-VMM marginally outperforming the TC-RBM. Finally, we evaluate the short order statistics of the models, using the Kullback-Leibler divergence between test sequences and model samples, and show that our proposed methods match the statistics of the music genre significantly better than the VMM.Comment: in Proceedings of the ECML-PKDD 2011. Lecture Notes in Computer Science, vol. 6913, pp. 289-304. Springer (2011

Linking vital rates of landbirds on a tropical island to rainfall and vegetation greenness

Remote tropical oceanic islands are of high conservation priority, and they are exemplified by range-restricted species with small global populations. Spatial and temporal patterns in rainfall and plant productivity may be important in driving dynamics of these species. Yet, little is known about environmental influences on population dynamics for most islands and species. Here we leveraged avian capture-recapture, rainfall, and remote-sensed habitat data (enhanced vegetation index [EVI]) to assess relationships between rainfall, vegetation greenness, and demographic rates (productivity, adult apparent survival) of three native bird species on Saipan, Northern Mariana Islands: rufous fantail (Rhipidura rufifrons), bridled white-eye (Zosterops conspicillatus), and golden white-eye (Cleptornis marchei). Rainfall was positively related to vegetation greenness at all but the highest rainfall levels. Temporal variation in greenness affected the productivity of each bird species in unique ways. Predicted productivity of rufous fantail was highest when dry and wet season greenness values were high relative to site-specific 5-year seasonal mean values (i.e., relative greenness); while the white-eye species had highest predicted productivity when relative greenness contrasted between wet and dry seasons. Survival of rufous fantail and bridled white eye was positively related to relative dry-season greenness and negatively related to relative wet-season greenness. Bridled white-eye survival also showed evidence of a positive response to overall greenness. Our results highlight the potentially important role of rainfall regimes in affecting population dynamics of species on oceanic tropical islands. Understanding linkages between rainfall, vegetation, and animal population dynamics will be critical for developing effective conservation strategies in this and other regions where the seasonal timing, extent, and variability of rainfall is expected to change in the coming decades