One of the main goals of lower division “service” mathematics courses is to provide STEM-intending students with opportunities to engage in activities and contexts that can support their efforts to apply the mathematical ideas they are learning to successive major courses. The Mathematics Association of America has supported many mathematicians’ efforts to ask partner discipline faculty what topics and habits of mind they feel should be covered in mathematics classes to prepare them for their subsequent classes. We add to this work with a twist: Instead of asking physics faculty what they want students to know, we analyzed videos that students in an introductory physics class created so that we could ask ourselves what mathematical practices were most and least prevalent in the students’ physics problem-solving efforts. A qualitative analysis of the results, which we present here, indicated that most students were proficient in math practices involving problem setup and that the majority were able to apply mathematical concepts such as trigonometry and the solving of algebraic equations. However, only 44% of the student groups concluded their explanations by discussing answer reasonability and only 18% conducted a unit analysis to determine if their answers were applicable to the context of the problems assigned, even though both of these elements are important components of the overall sensemaking process. This report presents examples that illustrate these results and concludes with implications for teaching both entry-level mathematics and physics courses by modeling productive problem-solving and sensemaking practices.

In this work, we generate optical vortices from the supercontinuum output of an ultrafast laser interacting with a micro-structured fiber. Using a segmented spatial light modulator, multiple vortices are designed and dynamically generated and shifted in order to observe their superposition in the image plane. It is shown that single-color patterns of exquisite complexity can be generated across a wide frequency range. Multi-color interference patterns are experimentally generated and compared to the results of computer simulations. Multiple vortices of varying colors are also generated and independently controlled, demonstrating that no spatial interference occurs. Experimental results are compared with theoretical and numerical simulations, showing excellent agreement.