Engaging in productive struggle during problem-solving activities can be viewed as critical to learning mathematics. In this paper, we explore how mathematical praxeologies can be utilized to describe students’ struggle with mathematical problem-solving. The results indicate that the students’ productive struggle with an optimization problem involving functions, graphs, and derivatives, contains long sequences where the students switch rapidly within and between mathematical praxeologies. During the work session as a whole, the students mix coherent arguments with tentative and on occasion unproductive arguments focusing on technology. One notable exception among these attempts is a theoretical moment where the students reduce the complexity of a previous subtask by deducing a general equation which provides them with key ideas on how to solve the given task.
Through the analysis of the different iterations of the Geometry Mobile (GEM) project, a mobile learning effort in the field of mathematics, we have identified a major architectural issue to be addressed in the design and implementation of m-learning applications. Due to the dynamic nature of the field many challenging requirements are continuously emerging. One of them relates to the possibility to support collaborative activities that demand sharing resources between students and their mobile devices in constantly changing conditions. These situations generate the need of using decentralized distributed architectures in which mobile devices can share resources to carry out the activity covering the concerns defined by the different stakeholders. This paper describes our current efforts connected to identifying a set of requirements for M-Learning activities. Thereafter, we elaborate on why a decentralized distributed system (DDS) can be used to provide a novel solution to tackle the mentioned above problems. Moreover, initial aspects related to the design of a DDS, including a self-adaptation mechanism are presented.
In this paper, we describe an approach for designing and developing technological solutions to support teachers in creating their own outdoor teaching activities. We elaborate on one particular case, TriGO, in which primary school students perform outdoor tasks to experience concepts and constructions in the field of mathematics. The application designs and an initial evaluation of the developed technological solutions is provided based on the results obtained from school activities performed with more than 10 teachers and 50 students.
In recent years, teaching mathematics in an outdoor setting has become popular among teachers, as it seems to offer alternative ways to motivate children’s learning. These new learning possibilities pose crucial questions regarding the nature of how mathematical activities should be designed for outdoors settings. In this paper we describe our current work related to the design and implementation of mathematical activities in this particular environment in which a specific mathematical content was used as the central component in the design. We illustrate our collaborative design approach and the results from observations of two activities. Our initial results provide us with valuable insights that can help to better understand how to design and implement this kind of educational activities.
The authors of this paper are involved in an ongoing project with the aim of investigating ICT-supported activities for the learning of mathematics where real-world images are mixed with computer-generated 3D images. The present paper explores the ways in which four students (15 years old) try to make sense of a task that calls for reflection on the concept of scale. The analysis shows how this specific kind of learning activity can challenge students to vary and coordinate among representations offered within the activity, thereby creating opportunities to extend and strengthen their networks of knowledge elements associated with the current learning object.
Students’ different learning performance on mathematical problem solving across contexts has attracted a number of researchers’ interest. The study investigates the spatial orientation ability of primary school students in an outdoor situation, where six pairs of grade six students are asked to coordinate themselves physically in terms of given distances with respect to two given points. Their spatial orientation performance is evaluated quantitatively, in terms of the number of attempts needed to reach the target points, as well as qualitatively by analyzing their strategies as described in their answers to a questionnaire. According to our findings the students enjoy and perform remarkably well in the outdoor setting, an observation that leads us to suggest that engaging students in outdoor activities may enhance their learning of mathematics.
Mathematical tasks in tests are central to students’ learning.Research shows that there is a significant gap between mathematical tasks in national tests and teacher-made tests.In this pilot study we examineeighteenSwedish and Chinese teachers’ viewsonwhat constitutes of a good mathematicaltesttaskat thelowersecondary school level. E-Mail Interviewing is conducted by presenting sevenmathematical tasks from national tests inSweden and China, respectively. The preliminary results show that Swedish and Chinese teachers hold some common viewsonthecharacteristics of good mathematical test tasks, but they also show different viewson the cognitively demanding tasks. Implications of the results and the methodology informed by the pilot study are discussed.
The unfolding of an affordance, as an opportunity for action, during a learningactivity requires the learner to interact with mediating artifacts. The design of alearning activity involves appropriating affordances and embedding them in theactivity in such a manner that the learner is invited to unfold the affordances,through interaction with their mediating artifacts in accordance with pre-definedhypothetical learning trajectories. In this paper, the notion of affordances is usedexplicitly in the discussion of two previous research efforts. We argue that thenotion of affordances, which was tacitly used in these efforts and aligns well withthe methodology of scenario-based design, may be used as an instrument for thecollaborative design of innovative mathematical learning activities.
In this paper we describe and reflect on the design of a mathematical learning activity developed incollaboration between teachers, researchers and technical developers. By making use of augmentedreality (AR) as a technology supporting augmentation of a real-world projection with computergene-rated images, we have designed an activity that promotes unique action and learningtrajectories. These trajectories require the learners to engage in interactive-constructive actions thatinvolve and stimulate the development of their self-regulatory skills by inviting them to vary andcoordinate across the contextual affordances of the technologies and the physical resources in theclassroom. Our learning activity is designed as a collaborative guided inquiry, implemented in aregular classroom and involved mathematical problem solving in relation to the geometric conceptof scale. In order to successfully complete the activity, the learners are challenged to coordinateaffordances from three distinct referential contexts by involving physical and virtual artifacts. In thedesign process, we identify critical aspects of the activity and embed affordances for correspondingscaffolding actions which turn out to play a crucial role when the activity is implemented with agroup of four 15-year-old students. Although the AR technology has served us well in developingthis particular activity, this specific technology appears to have limited applicability in mathematicseducation beyond geometry. We recommend that future research efforts move beyond AR andconsider the broader context of embodied design with tangible user interfaces, that have recentlyshown great potential for the design of innovative activities for the learning of mathematics.
Flera års forsknings- och utvecklingsarbete har resulterat i en handfull lärandeaktiviteter för elever i årskurs 4-9, som i undersökande aktiviteter använder ny teknik i form av datorer, projektorer, interaktiva skrivtavlor,webbkameror och mobiltelefoner. Tekniken kan låta oss möta meningsfull matematik på nya sätt i nyasammanhang och kan göra det möjligt att sammanfläta undersökande utomhusaktiviteter i små grupper medinomhusaktiviteter i helklass. Vi ger flera konkreta exempel på hur tekniken kan stimulera och stödjakommunikation och representation av ett matematiskt innehåll.
Guided by the notion of design research we develop a learning activity for 12 year old students, who are asked to coordinate themselves physically in terms of distances with respect to two given points in an outdoor setting. The outdoor activity, as well as its continuation into the mathematics classroom, involves mobile software applica-tions specifically developed to support this activity. In this paper, we argue that the design of innovative learning activities is enhanced by the coordination of expertise and knowledge from several research domains, whose collaboration is facilitated by using affordances for representation and communication as design instruments. We present a case where ancient Greek mathematics, modern psychology and techno-logical affordances guide the design of an innovative spatial coordination activity.
Three teachers and a researcher have co-designed a teaching activity intended tosupport students’ learning of two strategies for subtraction. The researcher hasput focus on the relation between theoretical principles, introduced to underpinthe participating teachers’ work, and the learning outcomes of their 33 studentsin grade 4. The principles are adapted by the researcher during three designcycles and negotiated with the teachers to meet emerging needs in the designprocess. The three teachers are fully responsible for planning, implementing, andevaluating an iterated teaching activity designed according to these principles.This study indicates positive effects of targeting low-achievers with teacher-ledstructured group activities, using guiding principles from self-regulation theory.
We design and evaluate a curriculum-based mathematical learning activity involvingsecondary students’ geometrical constructions, mathematical modeling and algebraicvalidation of hypotheses based on hands-on explorations with the interactive geometryapplication GeoGebra available on individual laptops. We argue that guided inquiries in atechnology-enhanced learning environment that invites blending of interactive technologiesand traditional resources may be an efficient means for developing self-regulatory skills.