We present the subalgebra structure of sl(3, O), a particular real form of e 6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3, O) to construct the multiplication table of the corresponding Lie algebra sl(3, O). Both the multiplication table and the group are then utilized to find various nested chains of subalgebras of sl(3, O), in which the corresponding Cartan subalgebras are also nested where possible. Because our construction involves the Lie group, we simultaneously obtain an explicit representation of the corresponding nested chains of subgroups of SL(3, O).

The process of complexification is used to classify a Lie algebra and identify its Cartan subalgebra. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to sl(3, O), a real form of e 6 with signature (52,26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within e 6 . Motivated by an explicit construction of sl(3, O), we then construct an abelian group of order 8 which acts on the real forms of e 6 , leading to the identification of 8 particular copies of the 5 real forms of e 6 , which can be distinguished by their relationship to the original copy of sl(3, O).

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Curriculum developers are interested in how to leverage various instructional tools -like whiteboards, Mathematica notebooks, and tangible models -to maximize learning. Instructional tools mediate student learning and different tools support learning differently. We are interested in understanding how the features of instructional tools influence student engagement during classroom activities and how to design activities to match tools with instructional goals. In this paper, we explore these questions by examining an instructional activity designed to help advanced undergraduate physics students understand and visualize the electrostatic potential. During the activity, students use three different tools: a whiteboard, a pre-programmed Mathematica notebook, and a 3D surface model of the electric potential. We discuss how the tools may be used to address the the instructional goals of the activity. We illustrate this discussion with examples from classroom video. I. RESEARCH QUESTIONS & METHODSedited by Ding, Traxler, and Cao; Peer-reviewed,

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Students need a robust understanding of the derivative for upper-division mathematics and science courses, including thinking about derivatives as ratios of small changes in multivariable and vector contexts. In Raising Calculus to the Surface activities, multivariable calculus students collaboratively discover properties of derivatives by using tangible tools to solve context-rich problems. In this paper, we present examples of student reasoning about derivatives during the first of a sequence of three Raising Calculus activities. In this sequence, students work collaboratively on dry-erasable surfaces (tangible graphs of functions of two variables) using an inclinometer, a tool that can measure derivatives in any direction on the surfaces, to invent procedures to determine derivatives in any direction. Since students are not given algebraic expressions for the underlying functions, they must coordinate conceptual and geometric notions of derivatives, building on their understandings from introductory differential calculus. We discuss examples of student reasoning that demonstrate how the activity supports student realization of the need to define a path, attend to direction and the orientation of the coordinate axes and recognize covariation between related quantities. This first activity enables students to initially recognize the ratio of small changes approach to derivatives. We briefly describe how students utilize the ratio of small changes approach in subsequent activities as they measure partial derivatives on surfaces (using an inclinometer) and on contour maps.