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and symbolic work. Technical challenges in science and engineering are getting more difficult, while dealing with numbers is getting easier. A consequence is that work on an abstract and symbolic level—even if only to organize numerical work—is increasingly important. But these skills are also declining. Some of my students have trouble with any problem whose answer is not a number: they can handle circles of radius 3, but simple problems with circles of radius ‘r’ are foreign territory. Again, I feel my students are better served if I can help develop these skills. My examples and problems usually have symbolic parameters, and I emphasize what these reveal about scaling, optimization, and error analysis. I usually use exact arithmetic. This preserves structure (π and √ 2 don’t disappear into decimals), and is half-way to symbolic work. Again this is a challenge, and quite a few students need remediation before it is accessible, but it is reasonable to expect them to get it before attempting a science and engineering curriculum. Applications. Applications provide opportunities for students to exercise their skills and see the methods in action. However applications do not have to be to physical problems, and in fact I find most physical applications unsatisfactory. • It is a good idea to plug in numbers from time to time but it destroys a lot of functionality. Printing out web pages can also be a good thing, but it kills the functionality of links. In particular, extended numerical applications do not exercise the most important skills. • Most of these students have specific interests. Applications that address their interests will duplicate material done in more depth in other courses. Applications that don’t address their interests don’t engage them. • Superficial applications are usually little more than vocabulary (replace ‘velocity’ with the first derivative, etc.). These are worth mentioning, but as testable material are not a good use of their time. On the other hand working a bit more abstractly and symbolically opens up mathematical topics to explore. These are, in effect, applications that are both missionrelated and quickly accessible because techniques and terminology are already in place. I also find that laying groundwork for mathematical applications helps organize and sharpen presentations. Resource constraints. Unfortunately, one more version should be addressed: “Even if we do figure out what students need, can we afford to provide it?” I individually, and the department at Virginia Tech collectively, have tried many things that improved learning but that had to be abandoned because they required unsustainable levels of faculty overtime. These include group projects along the lines

Doceamus: What Should Students Get from Calculus? (And How Can We provide It?)