Grounded and embodied cognition (GEC) serves as a framework to investigate mathematical reasoning for proof (reasoning that is logical, operative, and general), insight (gist), and intuition (snap judgment). Geometry is the branch of mathematics concerned with generalizable properties of shape and space. Mathematics experts (N ϭ 46) and nonexperts (N ϭ 44) were asked to judge the truth and to justify their judgments for four geometry conjectures. Videotaped interviews were transcribed and coded for occurrences of gestures and speech during the proof production process. Analyses provide empirical support for claims that geometry proof production is an embodied activity, even when controlling for math expertise, language use, and spatial ability. Dynamic depictive gestures portray generalizable properties of shape and space through enactment of transformational operations (e.g., dilation, skewing). Occurrence of dynamic depictive gestures and nondynamic depictive gestures are associated with proof performance, insight, and intuition, as hypothesized, over and above contributions of spoken language. Geometry knowledge for proof may be embodied and accessed and revealed through actions and the transformational speech utterances describing these actions. These findings have implications for instruction, assessment of embodied knowledge, and the design of educational technology to facilitate mathematical reasoning by promoting and tracking dynamic gesture production and transformational speech. Educational Impact and Implications StatementHow do mathematical intuitions arise, and how can they help with advanced forms of reasoning such as geometry proofs? One idea is that intuitions arise from body movements that allow people to directly experience mathematical ideas and relationships. We analyzed videotaped interviews of 46 mathematics experts and 44 nonexperts and found they are each more likely to show correct mathematical intuitions and generate mathematically valid proofs when they produced gestures while speaking. The research findings contribute to theories of embodied cognition by showing that people can tap into nonverbal ways of mathematical thinking. This work is important for education in STEM (science, technology, engineering, and mathematics) because it demonstrates that embodied cognition applies beyond basic mathematics such as counting and computation to conceptual forms of reasoning involved in geometry proofs.

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Do concrete learning materials promote strong learning outcomes, or do they simply make learning tasks more initially accessible? Although concrete materials may offer an intuitive foothold on a topic, research on desirable difficulties suggests that more challenging tasks facilitate greater retention and transfer. In the approach introduced here, grounded coordination challenges (GCCs) are embedded into the design of concrete learning materials to deliberately increase the difficulty of the learning task. More specifically, these challenges are intended to promote a deliberative process of mapping between perceptual elements of the materials. In 2 experiments the GCC approach was tested in a number line estimation task by comparing training with an "incongruent ruler"-which was designed to mismatch the length of an on-screen number line-to a "congruent ruler" (both experiments), or no ruler (the 1st experiment only). In both cases participants with the incongruent ruler were more likely to transfer knowledge to spatially transformed number lines. These results indicate that desirable difficulties facilitate learning in mathematical activities. Furthermore, the difficulties should emphasize a deliberate coordination process between critical features of the learning tool and the task. Implications for the design of learning activities that balance instructional support with conceptual challenge are discussed.

The study of children's creative thinking as they use technology to produce works of art, has only gotten humble attention in the literature (Loveless, 2002; Resnick, 2006). Although within the learning sciences, and in the field of new literacy studies, a growing number of compelling articles have made attempts to document the creative acts of young people using technology (Kafai, 2006; Sefton-Green, 2006; Gustavson, 2007; Squire, 2008), it is evident that there is little agreement on how to clearly conceive of this thing we call creativity. Quite often, when documenting the practices related to making things with technology, creativity is defined as the process of active engagement, some call it play (Lindqvist, 2003). Nevertheless, as we can identify through much of the data, not all children engage in the process of design and production in the same way, or to the same degree. This highlights how paramount the role and the thinking of the creative agent is to the act, and why it is in need of further investigation by psychologists, social researchers, designers and educators (Middleton, 2007).Additionally, it is vital to recognize that the technology plays a unique role in the process of creative expression (by scaffolding and supporting design with abstract representations) (Dede et. al, 1997; Bransford, 1999; Black, 2006). How then, can we say that all of these acts are creative? Furthermore, there is great and ever growing diversity of what new things individuals and groups actually make with technology -from video games, to blogs, to music. Some maintain that with the study of creativity in children all products that a child or youth produces should be considered creative (Amabile, 1996), despite acknowledging that creativity must be cultivated and developed. Can we assume that all the products young people produce with technology are creative?We will present findings from a study that undertook an examination of the iterative design process, and how certain cognitive skills relate to creativity. The study examines a group of 4 th grade boys, ages 9 -11, who participated in a twelve week after school video game design course at an elementary school in New York City. The course focused on the design of video games, with curriculum that introduced computer programming, mathematics, storytelling, and the digital arts. This effort was also grounded in the theory of media literacy, examining how video games and interactive texts extend the new literacy paradigm (Gee, 2003; Leu et. al, 2004). Both qualitative and quantitative data analysis was employed to examine artifacts produced by all subjects. Observational methods of research, including field notes, student interviews, were employed to examine the social relationships and the dynamic learning environment. Additionally, student work including storyboards, art and video games code, serve as a major source of data.