This aim of this study is to describe the process of local conjecturing in generalizing patterns based on Action, Process, Object, Schema (APOS) theory. The subjects were 16 grade 8 students from a junior high school. Data collection used Pattern Generalization Problem (PGP) and interviews. In the first stage, students completed PGP; in the second…

This study contributes a methodological tool to reconstruct the cognitive processes and mathematical activities carried out by mathematical modelers. Represented as Modeling Transition Diagrams (MTDs), individual modeling routes were constructed for four engineering undergraduate students. Findings stress the importance and limitations of using…

In 2009, we reported on a new Intelligent Tutoring Systems (ITS) technology, example-tracing tutors, that can be built without programming using the Cognitive Tutor Authoring Tools (CTAT). Creating example-tracing tutors was shown to be 4-8 times as cost-effective as estimates for ITS development from the literature. Since 2009, CTAT and its…

The desire to persuade students to avoid strictly memorizing formulas is a recurring theme throughout discussions of curriculum and problem solving. In combinatorics, a branch of discrete mathematics, problems can be easy to write--identify a few categories, add a few restrictions, specify an outcome--yet extremely challenging to solve. A lesson…

When a second-grade class struggled to make sense of the algorithms for multidigit addition and subtraction, attitude and morale were down, and student discourse was at an all-time low. When daily mathematics class started, the teacher was hearing an overall class moan. The instructed needed to turn it around, to find a way to help excite and…

Although most tasks that STEM professionals engage in--like identifying problems, making models, and testing those models--involve inquiry, many STEM classes still rely on direct instruction. Stephan argues that even as new resources for active learning are being developed for STEM instruction, many teachers aren't using these resources and tasks…

Recognizing and using mathematical structure are key components of mathematical reasoning. The authors believe that one productive way to support students' use of structure is by identifying opportunities to address structure in the context of what teachers are already doing, rather than developing additional tasks or new curriculum materials. The…

Introduction of a pedagogical concept, Kubus, in a problem oriented learning context--analysed within the framework of an activity system--indicates what might happen when offering tools tempting to influence and regulate students' learning approach and hereby neglecting the importance of existing habits and values. Introduction of this new…

In this qualitative research, we studied difficulties that undergraduate students face while learning the concept of inner product space. Participants were 35 first-year mathematics students from Yildiz Technical University in the 2011 and 2012 academic years. We asked participants to solve 5 inner product space questions. Data were jointly…

In partnership with both formal and informal learning institutions, researchers have been building a suite of online games, called choicelets, to serve as interactive assessments of learning skills, e.g. critical thinking or seeking feedback. Unlike more traditional assessments, which take a retrospective, knowledge-based view of learning,…

Intelligent behavior in everyday contexts may depend on both ability and an individual's disposition toward using that ability. Research into patterns of thinking has identified three logically distinct components necessary for dispositional behavior: ability, inclination, and sensitivity. Surprisingly, sensitivity appears to be the most common…

Design involves solving complex, ill-structured problems. Design tasks are consequently, appropriate contexts for children to exercise higher-order thinking and problem-solving skills. Although creating engaging and authentic design contexts for young children is difficult within the confines of traditional schooling, recently, game-design has…

You read in the paper this morning that people who take multivitamins have fewer heart attacks and are less likely to get cancer than people who don't. Does this information make you more likely to want to take multivitamins? To truly prepare students for life, schools need to teach them the critical thinking skills they need to answer questions…

Teaching leadership skills is an important goal for many schools, both within the curriculum and beyond it. The traditional definition of a leader has been based on a hierarchy, with levels, a chain of command, and a "top dog." Yet other definitions exist, and for girls and women a redefined view incorporates an alternative concept of…

Computational thinking (CT) parallels the core practices of science, technology, engineering, and mathematics (STEM) education and is believed to effectively support students' learning of science and math concepts. However, despite the synergies between CT and STEM education, integrating the two to support synergistic learning remains an important…