Proofs produced by secondary school students learning geometry in a dynamic computer environment

As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.     

Proofs produced by secondary school students learning geometry in a dynamic computer environment

As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.This revised version was published online in September 2005 with corrections to the Cover Date.     

High school geometry students' justification for their views of empirical evidence and mathematical proof

Concerns about the use of computer-aided empirical verification in geometry classes lead to an investigation of students' understandings of the similarities and differences between the measurement of examples and deductive proof. The study reports in-depth interviews with seventeen high school students from geometry classes which employed empirical evidence. The analysis focuses on students' reasons for viewing empirical evidence as proof and mathematical proof simply as evidence.     

Discernment of invariants in dynamic geometry environments

In this paper, we discuss discernment of invariants in dynamic geometry environments (DGE) based on a combined perspective that puts together the lens of variation and the maintaining dragging strategy developed previously by the authors. We interpret and describe a model of discerning invariants in DGE through types of variation awareness and simultaneity, and sensorimotor perception leading to awareness of dragging control. In this model, level-1 invariants and level-2 invariants are distinguished. We discuss the connection between these two levels of invariants through the concept of path that can play an important role during explorations in DGE, leading from discernment of level-1 invariants to discernment of level-2 invariants. The emergence of a path and the usefulness of the model will be illustrated by analysing two students’ DGE exploration episodes. We end the paper by discussing a possible pathway between the phenomenal world of DGE and the axiomatic world of Euclidean geometry by introducing a dragging exploration principle.     

超级画板在提高数学师范生教师素质中的应用

超级画板作为一款功能强大的动态几何软件应用于数学师范生的培养,可通过创设情境,降低难度,透视本质,动手实验,创建模型等方式帮助学生学习数学知识.超级画板能提高学生的作图能力、教学设计能力、解题能力、编程能力等各种教师职业技能,并能对学生进行数学美的教育,并健全数学观.     

Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An intervention-based study with two Chinese classrooms

We conducted an intervention-based study in secondary classrooms to explore whether the use of geometric transformations can help improve students’ ability in constructing auxiliary lines to solve geometric proof problems, especially high-level cognitive problems. A pre- and post-test quasi-experimental design was employed. The participants were 130 eighth-grade students in two classes with a comparable background that were taught by the same teacher. A two-week intervention was implemented in the experimental class aiming to help students learn how to use geometric transformations to draw auxiliary lines in solving geometric problems. The data were collected from a teacher interview, video-recordings of the intervention, and pre- and post-tests. The results revealed that there was a positive impact of using geometric transformations on the experimental students’ ability in solving high-level cognitive problems by adding auxiliary lines, though the impact on the students’ ability in solving general geometric problems as measured using the overall average scores was not statistically significant. Recommendations for future research are provided at the end of the article.     

KNOWLEDGE USE IN THE CONSTRUCTION OF GEOMETRY PROOF BY SRI LANKAN STUDENTS

Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students’ attempts at proof development: geometry content knowledge, general problem-solving skills and geometry reasoning skills. Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills.     

Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course

We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.     

Proofs through Exploration in Dynamic Geometry Environments

The recent development of powerful new technologies such as dynamic geometry software (DGS) with drag capability has made possible the continuous variation of geometric configurations and allows one to quickly and easily investigate whether particular conjectures are true or not. Because of the inductive nature of the DGS, the experimental-theoretical gap that exists in the acquisition and justification of geometrical knowledge becomes an important pedagogical concern. In this article we discuss the implications of the development of this new software for the teaching of proof and making proof meaningful to students. We describe how three prospective primary school teachers explored problems in geometry and how their constructions and conjectures led them see proofs in DGS. Constantinos Christou: Author for correspondence.     

Understanding dynamic behavior: Parent–Child relations in dynamic geometry environments

The sequential organization of actions necessary to construct a figure in a dynamic geometry environment (DGE) introduces an explicit order of construction. Such sequential organization produces what is, in effect, a hierarchy of dependences, as elements of the construction depend on something created earlier. This hierarchy of dependences is one of the main factors that determine the dynamic behavior (DB) within a DGE, and the order is often explicitly stated by terms such as parent and child. This article is a part of a larger study that examines various instruments developed by users when they use dragging. It addresses one aspect of dragging: the connection between dynamic behavior and the sequential order of construction. Junior high students and graduate students in mathematics education were asked to predict the DB of points that were part of a geometric construction they had executed using a DGE according to a given procedure, and to explain their predictions. The study reveals that while hierarchy in geometric constructions in a DGE is mirrored by the DB, user actions and perceptions of DB indicate that users often grasp a reverse hierarchy in which dragging a child affects its parent. The study reveals four categories of reverse-order predictions, and suggests that these may be caused by terms and knowledge built into paper-and-pencil geometry, which are part of the resources users bring to dragging. Examining various DGEs¹ in detail reveals that in some cases the DB is in fact compatible with the reverse-order predictions. The paper concludes with a brief discussion of some implications for learning activities and software design. This revised version was published online in July 2006 with corrections to the Cover Date.     

La dialectique ancien-nouveau dans l'intégration de Cabri-géomètre à l'école primaire

In this article, we study the conditions and constraints of the integration of the dynamic geometry software ‘Cabri’ in the teaching of geometry in ordinary primary school classes (10 years old pupils). We focus our attention on the way the dialectic between old and new is working during this integration, looking at the types of tasks and techniques proposed in class by the teachers. The ‘good equilibrium’ between old and new ways of doing appears as one of the main conditions of integration as it allows to reconcile innovating and usual activities in the everyday life of the class. This revised version was published online in July 2006 with corrections to the Cover Date.     

Understanding of structured problem solutions

Much of the material provided in science courses consists of sample solutions to problems. This study is concerned with how such structured material is understood, what is earmarked for storage, what remains stored and how useful this is for later ability to recreate the problem solution. Understanding was assumed to be the result of a cognitive process in which information about the material understood is stored in memory. Twelve S's were required to “think aloud” as they understood three problem solutions: the solution of the Missionaries and Cannibals problem, a geometry proof, and a plan for another geometry proof. Each solution was presented step by step as requested by S. Immediately after he had understood the solution S was asked to recall the “outstanding points.” A week later he was asked to repeat the recall and to reproduce the solution. A hand coding of the “thinking aloud” protocol was analysed by a computer programme. Patterns detected by the programme suggest that the understanding process of an S has some consistency for different tasks. An analysis of the recalls of outstanding points showed that the kinds of points best retained were the context of the problems and subproblems within the solution. In the Missionaries and Cannibals task subproblems were positions reached. S's who recalled positions rather than moves reproduced the solution faster. In the geometry tasks the subproblems were steps with two or more premises. S's who recalled more of these reproduced the solution faster. In the geometry proof these S's had more adequately processed these steps when they were attempting to understand the solution. The importance of the S's method for deciding when he understands is suggested.     

Primary school pupils' ability to see scientific problems in everyday phenomena

Conclusion Although the experiments reported here are only a beginning to the research needed, the results obtained so far suggest that some teachers and some curriculum planners have overlooked factors in their consideration of problem solving as a learning method in primary school science. There appear to be teachers who have rejected this approach prematurely because many of the problems children suggest when they are first introduced to this method of working are not sound starting points for investigation. Curriculum palnners, on the other hand, do not appear to have given serious consideration to the fact that some subject matter provides a better starting point for pupil problem solving than others. Further, there has been inadequate information available for teachers on the type of classroom situations and teacher behaviour which will maximize pupils' ability to see investigable science problems in everyday phenomena. The type of research reported here will be continued and expanded to provide a clearer picture of contexts in which the ability of pupils to see investigable scientific problems is maximized.     

Pivotal teaching moments in technology-intensive secondary geometry classrooms

This study investigates three teachers’ uses of a dynamic geometry program (The Geometer’s Sketchpad) in their high school geometry classes over a 2-year period. The researchers examine teachers’ actions and questions during pivotal teaching moments to characterize mathematics instruction that utilizes technology. Findings support an association between teacher–tool relationships, predominant teacher actions, and questioning.     

Fostering empirical examination after proof construction in secondary school geometry

In contrast to existing research that has typically addressed the process from example generation to proof construction, this study aims at enhancing empirical examination after proof construction leading to revision of statements and proofs in secondary school geometry. The term “empirical examination” refers to the use of examples or diagrams to investigate whether a statement is true or a proof is valid. Although empirical examination after proof construction is significant in school mathematics in terms of cultivating students’ critical thinking and achieving authentic mathematical practice, how this activity can be fostered remains unclear. This paper shows the strength of a particular kind of mathematical task, proof problems with diagrams, and teachers’ roles in implementing the tasks, by analysing two classroom-based interventions with students in the eighth and ninth grades. In the interventions, the tasks and the teachers’ actions successfully prompted the students to discover a case rejecting a proof and a case refuting a statement, modify the proof, properly restrict the domain of the statement by disclosing its hidden condition, and invent a more general statement that was true even for the refutation of the original statement.     

How do teachers’ approaches to geometric work relate to geometry students’ learning difficulties?

Various studies suggest that French students (grades 7 to 10) may solve geometric problems within a paradigmatic framework that differs from that assumed by teachers, a situation prone to misunderstandings. In this paper, we study the extent to which secondary school teachers recognise the conflicting paradigms and how they handle the geometric work conducted, in sometimes unintended ways, by their students. This is done by analysing teachers’ reactions to specific answers students offered to the Charlotte and Marie problem, an “ambiguous” problem with various solutions depending on the paradigm adopted. As a result of the study, we found that, beyond similarities due to a shared mathematical background, the way secondary schoolteachers handle students’ answers varies with their conceptions of geometric work. Implications are drawn regarding the teaching of geometry and the training of teachers.     

Relating to the ‘Other’: transformative,intercultural learning in post-colonial contexts

In this paper we use Transformative Learning Theory as a lens for making sense of teachers’ learning from study visits to the Global South. Transformative Learning theory is made up of two main elements: the form of transformations and the processes that support transformations. ‘Life changing’ experiences as expressed by study visit participants have been interpreted as transformational, but questions about who and what are transformed, and whether this is at the expense of the ‘Other’, are rarely addressed. Drawing on data from a project investigating study visits for UK teachers to Gambia and Southern India, we analyse the form that changes take and discuss whether these can be seen as transformational. We argue that without an explicit focus on relational forms of knowledge about culture and identity, self and other, the potential for transformations in how we relate to, and learn from, each other in postcolonial contexts is severely diminished.     

Algebraic methods in plane geometry

In this article we examine the role of mappings in elementary geometry. After making some comments about the Erlangen programme initiated by Felix Klein in 1872, wherein he proposed a way of studying geometries based on the underlying transformation groups, we see how theorems like Von Aubel’s theorem and Napoleon’s theorem can be proved in an elegant manner using similarity mappings, and how some construction problems may be solved using isometries. At the end we present a recent proof by Alain Connes of “Morley’s Miracle”, based on affine transformations. Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject.     

Teachers' democratic and efficacy beliefs and styles of coping with behavioural problems of pupils with special needs

This study examines how teachers actually cope with behavioural problems of included students. In order to understand the impact of individual differences on teachers' coping strategies, the authors looked at the relationship between these strategies and teachers' democratic beliefs and self‐efficacy. Participants were 33 teachers in Israel, who teach inclusive classes (1st–3rd grade). Data were collected through classroom observations, teacher interviews and questionnaires. In the interviews, teachers reported that they preferred helpful strategies as a solution to behavioural problems; however, classroom observation revealed that teachers actually used more restrictive responses. These results indicated that there is a gap between teachers' hypothetical knowledge and their applications of this knowledge in authentic classroom situations. In addition, positive correlations were found between teachers' democratic beliefs, teacher efficacy and the use of helpful strategies in regard to different behavioural problems. Practical implications for teacher education are discussed.     

Tracking instructional quality across secondary mathematics and English Language Arts classes

Teachers have the largest school-based influence on student learning, yet there is little research on how instructional practice is systematically distributed within tracking systems. We examine whether teaching practice varies significantly across track levels and, if so, which aspects of instructional practice differ systematically. Using multilevel modeling, we find that teachers of low track classrooms provided significantly less emotional support, organizational support, and instructional support to students in their classes than did teachers of high track classrooms. Mathematics classes were also observed to have higher quality instructional support for both content understanding and analysis and problem solving than English classes. We develop cases illustrating how small but significant differences in instructional quality are associated with substantially diverging lived experiences for students in high and low track classes.