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Following Felix Klein, an advanced historical standpoint is here presented for teaching college geometry for teachers. Three main ways of developing an advanced historical standpoint are discussed with classroom experiments. One is building connections among geometries by developing an inquiry into definitions of geometric objects such as rhombus, their extensibility with their family relationships across Euclidean and non-Euclidean geometries. Second is on the multiplicity and extensibility of transformations as represented by two historical approaches advocated by Klein and Usiskin. The third way to develop an advanced standpoint is by developing a critical look into a geometry practice tracing its change with the reforms in school geometry. The practice of constructions to connect geometry and algebra is impacted by two historical efforts. One is a supportive effort by Hilbert on the practice of constructions by Hilbert’s Algebra of Segments dating back to 1902 to connect geometry and algebra. The other historical reform effort is by School Mathematics Study Groups (SMSG) during 1960s, which led to weakening the axiomatic foundations of the practice of constructability and exactness. The case of SMSG’s angle construction axiom is criticized in their revision of axiomatic foundations of school geometry. Three approaches to develop an advanced standpoint informing research and practice of geometry teacher education towards a more historically connected stance. 

Fourier analysis learning trajectories are investigated in this full paper as a joint interdisciplinary construct for a scholarly collaboration among engineering and mathematics faculty. This is a dynamic and recursive construct for aligning, developing, and sharing research based innovative practices for engineering mathematics education. Towards building more coherence and transfer of learning between engineering and mathematics courses, these trajectories offer experimental practice templates for the interdisciplinary community of practice for engineering mathematics education. Conjectured learning trajectories for Fourier analysis thinking are here articulated and experimented in three courses - Trigonometry, Linear Algebra, and Signal Processing. Informed by the interdisciplinary perspectives from the team, these trajectories help to design instruction to support the complex learning of the mathematical, and engineering foundations for the advanced mathematical concepts and practices such as Fourier Analysis for engineers. The re- sults highlight the impact of collaborative, interdisciplinary, and innovative practices within and across courses to purposefully build and refine instruction to foster coherence and transfer with learning trajectories across mathematics and engineering courses for engineering majors. This offers a transformative process towards an interdisciplinary engineering mathematics education. The valid assessment and measurement of complex learning outcomes along learning trajectories are discussed for engineering mathematics education, paving the pathway for our future research direction.