We suggest a possible explanation for this phenomenon and discuss its importance for teaching. A routine test of associated algebra and calculus skills indicated that many of the students were familiar with the key calculus concepts for solving these non-routine problems nonetheless, students often used sophisticated algebraic methods rather than calculus in approaching the non-routine problems. More than half of these students were unable to solve even one problem and more than a third made no substantial progress toward any solution. This paper reports on a similar study, using the same non-routine first-year differential calculus problems, with students who had completed one and one-half years of traditional calculus and were in the midst of an ordinary differential equations course. The abstract is as follows: In two previous studies we investigated the non-routine problem- solving abilities of students just finishing their first year of a traditionally taught calculus sequence. This article appeared in Research in Collegiate Mathematics Education,IV in the series, Issues in Mathematics Education, Vol.
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