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Students fresh out of high school are often under the impression that mathematics is a discipline based entirely in recognizing the type of problem and applying an algorithm or cookbook method. These students think there are finitely many types of problems, and each type of problem corresponds to a method. From this perspective, "doing math" is about, in order,

- Identifying the type of the problem
**Remember**ing the method that corresponds to the problem- Executing the method

However, one of the main goals of quality math courses is to carry students up to the next level of Bloom's Taxonomy, where they might **comprehend** instead of just **remember** what is going on. In fact, we hope they will comprehend strategies or facts that transcend specific "types" of remembered problems. I always hope to chip away at the idea of every problem being classifiable into memorized and cookbook categories.

I am interested in which topics in College Algebra or Precalculus best **reward** a student who ventures up a level in Bloom's Taxonomy in this way. These should be topics where a cookbook/**remember**ing student will be able to solve the problem, but a **comprehend**ing student will be able to solve the problem more easily.

14I don't think the problem is patterns as much as relying on algorithms/cookbook methods to solve problems. I had a student who would always solve simultaneous equations by substitution no matter how messy the algebra, even when it was clear (to me) that adding and subtracting the equations would be much easier in eliminating one variable. I would like to suggest that you look for problems that are exceptions to algorithms and not worry about patterns. – Amy B – 2018-01-24T10:41:13.503

6@AmyB you are absolutely right, pattern-recognition is really not my issue. I should have titled the question "Math topics that reward going beyond cookbook methods" or something similar. – Chris Cunningham – 2018-01-24T17:03:24.817

6You can still edit it and change the title! – Amy B – 2018-01-24T17:31:21.427

4A singular example – svavil – 2018-01-24T18:26:25.160

2I think pattern recognition and analogy are highly useful in both math and more applied subjects. Certainly much math research still relies on induction and analogy to get good hypotheses even if rigourous proof then shows it right or wrong. Any topic is more easily learned if organized into a structure. What is your heartache versus patterns and aren't there bigger issues with students to fix than pattern seeking? – guest – 2018-01-24T21:40:00.140

If you assume that the description of a problem fits one A4 page (or one entire book, or one 4TB HDD, ...), there are indeed finitely many types of problems. – Eric Duminil – 2018-01-26T07:51:18.163

@AmyB Thanks for the edit! – Chris Cunningham – 2018-01-26T15:29:07.250

Good edit to change "pattern recognition" to "cookbook...", considering that many people (myself included) think that pattern recognition, at various levels of course, is a big part of mathematical thinking (if not of the static body of established mathematics...) – paul garrett – 2018-01-27T23:35:21.750