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Some students report getting lost in the activity of abstraction. Two properties were associated with the idea of a solution to a problem: specificity of the problem and relation between the problem being solved, and any problem the students had encountered or anticipated encountering. Some students expressed a preference for solutions to specific problems. Some students express a preference for concrete problems. "I find it easier to work with things that are concrete rather than abstract. " Their immediate concern was to learn to solve a specific problem, before attending to a technique for solving more generally, i.e., solving problems of that type. Some students remark that they have noticed they are better prepared than other students to attend to a technique for a class of problems rather than a specific problem. These more able students explain that they know that a technique can be applied to another specific problem. Some students found pointing out analogies, where one problem instance had commonality with another instance, helpful. Some students observe that analogies are insufficient. In particular, students learning traveling salesman problems might not recognize these as more general problems on graphs. Some students observe that proof writing typically occurred at a level of abstraction higher than a single concrete instance, and that this made the material more difficult. Other students seemed to miss the import of operation at a higher level of abstraction, not seeing the multiplicity of instances of a general notion: "With recursion you can apply the principle to many different algorithms, trees, here, there, everywhere, but proofs, that's it. It's absolute and true you accept it and move on." and "face the fact that a lot of proof writing doesn't have anything to do with a specific problem, a specific circumstance". Combined with the difficulties we see students facing with mathematical formulation, one might consider the possibility that some students might be taking a superficial approach, manipulating formulation without attaching meaning at a lower level of abstraction. So, we have some students operating in concrete but not abstract, we might also have students operating at abstract but not concrete. If this were true it would not be surprising that some students feel anxiety or alarm at having to work with proof. "They gave us sequences and series and summations and they're like, write the equation for this and they were awful." |