Is it worth it to understand why math works, even in college? Or because of the pace should I just stick to memorizing steps and formulas.

Question

I will be taking a college pre-calc math class soon and I was wondering if it is worth to try to understand math or just memorize formulas. You know, all the "Why does this work?" questions and math history. I do have an interest in learning how proofs and formulas work but I won't do it if it affects my grade.

Answer 1

This is to some extent a matter of what your opinion is on the purpose of education.

Nevertheless, I think that, ideally, the choice you describe is not possible—a university class that allows students to pass merely by cramming prescriptions for solving problems in lieu of flexing higher-order thinking skills is not a university-level class at all.

An ideal university-level mathematics course provides a setting to nurture and demonstrate critical thinking skills—if all you have to do to pass is "memorize formulas", somebody isn't doing their job!

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But what skills does mathematical critical thinking comprise? Adapting from Arons's (1997) monograph on teaching college physics, demonstrating critical thinking entails

1. consciously demanding proofs or derivations for supposed results;
2. recognizing gaps in proofs and derivations;
3. discriminating between the truth of conjectures in general and their truth in special cases;
4. requiring prior agreement on the definition of technical terms before accepting their use;
5. unearthing implicit assumptions in proofs and derivations;
6. drawing sound inferences, in particular
   • making use of syllogistic reasoning to derive theorems from hypotheses and
   • using computational evidence to formulate conjectures;
7. performing hypothetico-deductive reasoning, i.e., synthesizing existing knowledge and applying it to unfamiliar systems to come up with plausible ideas concerning how they work;
8. discriminating between inductive (i.e., statistical) and deductive (i.e., logical) arguments;
9. testing one's own reasoning and conclusions for internal consistency, and
10. consciously scrutinizing one's own cognitive processes ("What is my way of doing mathematics?").