Working Backwards from and Vertically Integrating within the Capstone
The Capstone is the culminating course in our Operations Research and Computer Analysis (ORCA) undergraduate major. Students work in small teams on applied O.R./Analytics problems. (See this page for links to ones we have done pertaining to sustainability.) Some of the outcomes we have for students through this course are:
- define, formulate and solve an O.R./analytics problem
- use technology and software to aid in problem solving
- read/understand mathematical texts/papers
- write sound technical passages
- prepare and deliver an oral presentation
- work as an effective member of a team
The idea is that these outcomes and others are progressively targeted over the prior 3 1/2 years across the courses leading up to the Capstone. In this way, the students are ready to achieve this final set of culminating outcomes in the Capstone course. And in reality, we do target many goals in earlier courses that support this development. For example, modeling is emphasized in courses such linear optimization, network & nonlinear optimization, decision models, probability models, linear regression, and more. Technical writing and oral presentation projects pop up in various courses along the way. In a program review effort a few years ago, we documented the overall departmental outcomes and the extent to which each was addressed in each course. In an even earlier review in 2000, a matrix of outcomes was created giving a quick visual picture of where the outcomes are addressed over time and how they build. We are due for a review and more refined look at this outcome development.
At Wooster College, seniors undertake an “Independent Study” (IS) in which they come up with a project and are mentored on it by faculty. The whole four-year academic experience at Wooster is geared towards the IS. According to a description in the book Colleges That Change Lives:
Professors infuse every course with a set of skills students must master to be successful researchers: They learn to frame questions, identify reliable sources, analyze primary documents, practice methodologies and write cogently – precisely the skills that are invaluable in the workplace and graduate school. In essence, the curriculum is built backward, starting with a clear vision of what students need to complete IS successfully. That vision guides how professors teach even introductory courses.
The concept of building the curriculum backwards makes a lot of sense to me and we have started having conversations about doing this in our department. In some cases, some of us have noted shortcomings in our students at the Capstone level and tried to address these earlier on in the curriculum in more of an ad hoc fashion. For example, taking a situation that could possibly be modeled a certain way, but that is not presented neatly in textbook form, and bringing it to a formulated model has sometimes been a trouble spot for some students. We have started to incorporate assignments in earlier courses to help address this. The scaffolded project-based learning scheme I spoke about in this webinar (using sustainability as the subject matter) has been a useful paradigm for this effort. We can provide the methodology to be used to the students (e.g., linear programming) but have them supply problem situation complete with parameters, objective, and constraints. On the other hand, late in the curriculum, we can provide a problem with any necessary data, but rely on them to look through their toolboxes for the appropriate methodology. I believe these efforts are beneficial but that they could be even more beneficial if undertaken in a systematic way so as to develop the desired Capstone competencies incrementally throughout the curriculum.
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In the book The Mind at Work by Mike Rose, there is a neat passage about a high school woodshop class having students ranging from freshman through seniors that ran like a kind of guild. The teacher was the master craftsman while the seniors had developed an impressive set of skills and experience and could build and/or supervise fairly extensive projects. The freshman were learning the basics, getting hands-on training, absorbing information and concepts from the upper class. Between was a continuum of learning with younger learning from older. I think this would be an effective way to run an O.R. Capstone as well. As they do now, the projects would start with a situation. The students of all the levels wrestle with the intangibles trying to refine it to a problem statement. The upper class familiar with methodology begin to cast the problem in those terms. Mid-level students can do the formulating and start the solving, analysis, coding, etc. under the supervision of the seniors, with the lowest level students pitching in on some of the lower level mechanics involved in these processes. Then all the students work on solution interpretation and an iterative process ensues in which students are involved to varying degrees depending on what their level of training is, but also based on what other things they bring to the table – creativity, hard work, grit, gumption, etc. This could probably be applied in other areas besides the Capstone such as across the core math courses (a differential equations project is going to require skills from pre-calc, calc I, and calc II – so put students from those four courses together on a team), and across the STEAM disciplines, if not beyond. And/or maybe the faculty serve as team members doing work on the project, not just advising. I did a quick google search for this type of situation and came upon “vertically integrated projects”. Georgia Tech has such a program that is multidisciplinary, vertically-integrated (sophomores through PhD students), and long-term. The long-term aspect is another potential bonus in that students can continue work on a large project from year to year but at a deeper level each year. I plan to look more into this notion of vertically-integrated projects and hope to help implement something like it at some point.