In my Keynote Address to PME-NA 45, I offer an embodied framework for naming what makes mathematics powerful for mathematicians and scientists, yet intractable for many learners. The essential claim is this: Students reside in the Real World, where math is grounded, embodied and meaningful, while mathematics resides in the ungrounded, disembodied…

Past efforts to teach novices programming through pair programming and project-based learning utilizing different low floors, high ceilings and wide walls platforms have been successful. Building from related work, this study investigates the effectiveness of Productive Failure (PF) pedagogical design in supporting youth and novices when learning…

Mathematics is not seen as an absolute corpus of well-founded knowledge anymore, but as a human activity, a social phenomenon, part of human culture that evolves historically and intelligibly in a social context. By harnessing the indigenous cultures and their knowledge systems in-depth, this article aims to evaluate how the sustainable…

This paper proposes that given today's "wicked" problems (Malhotra, 1997), where there are complex issues with no clear answers, where boundaries are fuzzy and the outcome is usually never known and unexpected, creativity can be enhanced, at appropriate moments, by making modes of thinking explicit. Using a particular heuristic approach…

Students need tools, thinking skills, to help them think actively and in depth about biological phenomena. They need to know what kind of questions to ask and how to find answers to those questions. In this article we present a toolkit with 12 "thinking tools" for asking and answering questions about biological phenomena from different…

Well-known historical accounts of questioning used in the education process trace back to Socrates. One of the best examples of his use of questioning is found in Plato's classic work "The Republic" (2003). Today, teachers still use questions as one way to help students develop productive thinking skills and to understand concepts and topics.…

Subjects for which aesthetics and creative performance are critical curricular dimensions (such as art, physical education, music, and technology education), and which are accommodative of students across the range of intelligences (Gardner, 1999) are not readily or completely captured by content standards. Therefore content knowledge in these…

Pullout pages provide suggestions for teaching elementary students at all levels about the wonders of human ingenuity. The suggestions help students see that great ideas come from the need to solve real-life problems and that one great idea leads to another, so current inventions help predict future inventions. (SM)

This paper describes how secondary teachers can use the Purdue Three-Stage Model as a framework for instruction that facilitates talent development in both heterogenous and homogenous classrooms. In the first two stages, students master the core content of a discipline and practice thinking creatively and critically about the content. In the third…

Argues that students should continuously question the purpose of their daily classroom activities and assignments. Explains that when students are required to consider purpose, they begin to develop important intellectual skills, and to bring relevance to their critical thinking and daily life. (NB)

The book "Productive Thinking", by Gestalt psychologist Max Wertheimer, was published over 50 years ago, yet it continues to wield influence. The book's argument on productive thinking, and why it is still relevant today, are the focus of this paper. Productive thinking involves going from a situation of bewilderment or confusion about some issue…

Describes a study conducted at an 8000-student Catholic university to determine whether critical thinking (CT) skills increase after four years of undergraduate education. Examines seven categories of CT: (1) truth-seeking; (2) open-mindedness; (3) analyticity; (4) systematicity; (5) CT self-confidence; (6) inquisitiveness; and (7) maturity of…

A review of the critical thinking literature reveals that there is growing interest in developing instructional strategies to develop students' critical thinking skills and dispositions. This paper reviews the research literature regarding the nature of critical thinking in the following areas: (1) differences in the conceptualization of critical…

This paper presents the preliminary results of a longitudinal study to assess the development of critical thinking as preservice teachers progress through their educational program. Students will be assessed during their final year in the program to help give an overview of the growth of their critical thinking skills. The Cornell Critical…

This paper demonstrates a variety of activities designed to enrich the learning environment for all children through the use of divergent thinking activities. The first activity involves structural indexing, whereby students brainstorm words found in a reading selection and construct sentences using a bingo formation, with the concentration on…