throughout science and engineering. For example, this approach can be facilitated by the development of a number of examples that explain uncertainties associated with natural phenomena and engineering systems (e.g., the ball-drop examples in Chapters 1 and 5). This step can be followed by an introduction to probabilistic thinking, including classical as well as Bayesian statistics. Many of these ideas can likely be integrated into existing courses rather than requiring the introduction of new courses into an already-crowded curriculum. The engineering design process, as embodied in capstone design courses, can then be presented as a decision process aimed at selecting from competing alternatives, subject to various constraints. This formulation of the design process has the added benefit of articulating to the student the scientific distinctions between various design paradigms or procedures (usually presented in the form of design recipes). It may be that some programs already have such approaches, but they are not common.

•  It is important to teach students to regularly confront uncertainty in input data and corresponding uncertainty in their stated answers. The committee encourages instructors in traditional courses to pose questions that include uncertainty in the input formation.

•  Similar to engineers and scientists, students of probability and statistics should acquire training in mathematical modeling as well as in computational and numerical methods. Again, the path to doing so should build on the logical sequence of discipline-specific core training. The key is understanding how probabilistic thinking fits into the scientific process (e.g., how probability fits together with mathematical modeling) and also understanding the limits of computation.

Recommendation: The elements of probabilistic thinking, physical-systems modeling, and numerical methods and computing should become standard parts of the respective core curricula for scientists, engineers, and statisticians.

•  Programs in management sciences. The intellectual framework represented by VVUQ seeks to assess the uncertainty in answers to a problem with respect to uncertainties in the given information. It is unlikely that students who are being trained as policy makers are going to be routinely interested in computational modeling, but it is important that they be educated in assessing the quality and reliability of the information that they are using to make decisions and also in assessing the inferential limits of the information. In the VVUQ context, this can mean, for example, understanding whether or not to trust a model that has undergone a VVUQ process, or understanding the distinction between predictions that have been informed by observations and those that have not.

It will be a challenge for individual university departments to take the lead in integrating VVUQ into their curricula. An efficient way of doing so would be to share the load among the relevant units. A way forward is emerging as a result of the DOE’s Predictive Science Academic Alliance Program (PSAAP). For example, at the University of Michigan’s Center for Radiative Shock Hydrodynamics (CRASH), both graduate and undergraduate students are included in the fundamental VVUQ steps as part of CRASH’s core mission. More importantly in the current context, as a result of PSAAP, the university is initiating an interdisciplinary Ph.D. program in predictive science and engineering. Students in the program have a home department but will also take courses and develop methodology relating to VVUQ. (A course in VVUQ has already been taught.) The computational science, engineering, and mathematics program at the Institute for Computational and Engineering Sciences at the University of Texas also has a similar graduate program. It is not hard to imagine a similar interdisciplinary program (perhaps a certificate program in predictive science) being rolled out to undergraduate students in engineering, physics, probability and statistics, and possibly management science.

Finding: Interdisciplinary programs incorporating VVUQ methodology are emerging as a result of investment by granting bodies.

Recommendation: Support for interdisciplinary programs in predictive science, including VVUQ, should be made available for education and training to produce personnel who are highly qualified in VVUQ methods.



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