Proof image

The emergence of a proof image is often an important stage in a learner’s construction of a proof. In this paper, we introduce, characterize, and exemplify the notion of proof image. We also investigate how proof images emerge. Our approach starts from the learner’s efforts to construct a justification without (or before) attempting any formal argument, and it focuses on the process by which a complete but not necessarily communicable image of that justification becomes available to the learner and provides explanation with certainty. We consider the interplay between the learner’s intuitive and logical thinking and, using the theoretical framework of Abstraction in Context, we trace the construction of knowledge that results from and enables progress of this interplay. The existence and identification of proof images and the nature of the processes by which they emerge constitute the theoretical contribution of this paper. Its practical value lies in the empirical analyses of these processes and in the potential to apply them to the design of tasks that support students in constructing their own proofs images and proofs. We believe that such processes are likely to considerably enrich students’ mathematical experience.     

Upper Secondary Schools and Sparsity: The Case of Northern Sweden

Governmental reforms since the early 1990s have acted as a push factor for starting new schools in Sweden and have created options for pupils to choose a school irrespective of municipal borders. This study highlights the upper secondary school system in small municipalities in northern Sweden since the late 1990s. The analysis shows that the reforms led to more schools, and thereby also to more competition in recruiting pupils both within and across municipal borders. When the cohorts started declining, the vulnerability of the school structure became obvious. The competition generated higher costs per capita and fewer available programs in small municipalities. Furthermore, it was noted that fewer pupils attending schools in small municipalities completed their education compared to those in larger municipalities.     

Secondary teachers’ knowledge of elementary number theory proofs: the case of general-cover proofs

In light of recent reform recommendations, teachers are expected to turn proofs and proving into an ongoing component of their classroom practice. Two questions emerging from this requirement are: Is the mathematical knowledge of high school teachers sufficient to prove various kinds of statements? Does teachers’ knowledge allow them to determine the validity of an argument made by their students? The results of this study, in which 50 secondary school teachers participated, point to a positive answer to the first question in the framework of elementary number theory (ENT). However, the picture is more complex with respect to the second one. Results indicated that some teachers may over-value the generality of symbolic mode of representation and under-value the generality of verbal ones. Possibly, the verbal representation of an argument is less transparent and more difficult to understand.     

Bridging the real and ideal: A comparison between learning community characteristics and a school-based case study

While a growing body of literature advocates the importance of school-based, collaborative action research, less attention has been focused on how these projects are developed and implemented in the early stages. This study, therefore, examines a small Canadian school's initial attempt at promoting a “learning community” approach and compares it to the ideals of collaborative teamwork set out by recent scholarship. Our findings suggest that collegiality holds promise for change in teachers' practice and meaningful professional development if the school's administrator accepts the learning community philosophy and is willing to take the lead in initiating the reform. As well, we found that all ideal characteristics set out in the literature need not be in place from the outset for a project of this sort to be successful.     

The Effectiveness of a Large‐Scale Language and Preliteracy Intervention: The SPELL Randomized Controlled Trial in Denmark

The present article reports results of a real‐world effectiveness trial conducted in Denmark with six thousand four hundred eighty‐three 3‐ to 6‐year‐olds designed to improve children's language and preliteracy skills. Children in 144 child cares were assigned to a control condition or one of three planned variations of a 20‐week storybook‐based intervention: a base intervention and two enhanced versions featuring extended professional development for educators or a home‐based program for parents. Pre‐ to posttest comparisons revealed a significant impact of all three interventions for preliteracy skills (= .21–.27) but not language skills (d = .04–.16), with little differentiation among the three variations. Fidelity, indexed by number of lessons delivered, was a significant predictor of most outcomes. Implications for real‐world research and practice are considered.     

倘若她是天使

有着俊朗外貌的Tommy Shane Steinerr以纯正的乡村歌喉演唱着属于他牛仔风格的歌曲。他出道后,受到了许多前辈的关注和提携:连续11年主持乡村音乐协会年度颁奖典礼的歌星Vince Gill出人意料地为Steiner的主推曲What If She’s An Angel配唱了合     

New actions upon old objects: A new ontological perspective on functions

We argue in this paper that the systematic use of special software in instruction has a profound impact on the notion of function as an abstract entity to be constructed. We argue that through the medium of the computer, the objects in the graphical, tabular and algebraic settings can change their essence and thus become objects of a new kind we callrepresentatives. Actions on representatives which naturally arise in this framework induce an ontological shift. A taxonomy of the skills involved in the learning of the concept of function through these new ontological lenses is presented, as well as software, and problem solving tasks that embody the same ontological perspective. Within the framework of a teaching experiment, students' acquisition of many of the identified skills was investigated by means of a questionnaire and interviews during computer supported problem solving sessions. The most salient results of the study indicate that a majority of students were able (1) to cope with partial data about functions (e.g., problems of interpolation and arbitrariness), (2) to recognize invariants (i. e., properties of functions) while coordinating actions among representatives from different settings, and (3) to recognize invariants while creating and comparing different representatives from the same setting.