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…

In the United States, students' learning experiences around proof is generally concentrated in the domain of high school geometry with a focus on its verification function. Thus, providing students with a limited conception of what proof entails and the role that it plays in performing mathematics. Moreover, there is a lack of U.S.-based studies…

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five…

This study explores the effects of the "presence of external representations of a mathematical object" (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric…

Mathematics is an important discipline, providing crucial tools, such as problem solving, to improve our cognitive abilities. In order to solve a problem, it is better to envision and represent through multiple means. Multiple representations can help a person to redefine a problem with his/her own words in that envisioning process. Dynamic and…

In CSCL research, collaborative activity is conceptualized along various yet intertwined dimensions. When functioning within these multiple dimensions, participants make use of several resources, which can be social or content-related (and sometimes temporal) in nature. It is the effective coordination of these resources that appears to…

The title is that of a course with the same name, designed for teachers of mathematics. The rational for a course specifically on geometry was that "many of those currently teaching mathematics in school had little geometrical education". Teachers on the course experience geometry through problem solving, and learning to pose geometrical problems.…

Many delegates at "conference" relish the opportunity, and the space, to "do some mathematics". Opportunity and space help to make the experience memorable, but how often is the quality of the starting point, or question acknowledged? Here is a set of starting points or problems that invite the reader to "do some mathematics". Deliberately, no…

Geometry provides a natural window into what it is like to do mathematics. In the world of geometry, playful experimentation is often more fruitful than following a procedure, and logic plus a few axioms can open new worlds. Nonetheless, teaching a geometry course in a way that combines both rigor and play can be difficult. Many geometry courses…

The emergence of non-Euclidean geometries in the 19th century rocked the foundations of mathematical knowledge and certainty. The tremors can still be felt in undergraduate mathematics today where encounters with non-Euclidean geometry are novel and often shocking to students. Because of its divergence from ordinary and comfortable notions of…

This paper describes how the author's students (in-service and pre-service secondary mathematics teachers) enrolled in college geometry courses use the Geometers' Sketchpad (GSP) to gain insight to formulate, confirm, test, and refine conjectures to solve the classical airport problem for triangles. The students are then provided with strategic…

Sometimes a student's unexpected solution turns a routine classroom task into a real problem, one that the teacher cannot resolve right away. Although not knowing the answer can be uncomfortable for a teacher, these moments of uncertainty are also an opportunity to model authentic problem solving. This article describes such a moment in Zahner's…

Background: The PISA study (assessing the competences of 15-year-old students in their mother tongue, in mathematics and in natural sciences) deals with the benchmarking of the OECD member states when it comes to meeting globally defined educational standards. Transnational educational experts now form the most powerful group within the…

The availability of sophisticated computer programs such as "Wolfram Alpha" has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is…

We examined geometric calculation with number tasks used within a unit of geometry instruction in a Taiwanese classroom, identifying the source of each task used in classroom instruction and analyzing the cognitive complexity of each task with respect to 2 distinct features: diagram complexity and problem-solving complexity. We found that…