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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.

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    $\begingroup$ Yes it is worthwhile. It is exercise for the mind. $\endgroup$ – copper.hat Sep 22 at 2:34
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    $\begingroup$ You should absolutely try to understand math and definitely should not merely memorize formulas. This is how one gets good at math, and the only way to enjoy math. $\endgroup$ – littleO Sep 22 at 2:37
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    $\begingroup$ My own point of view: the more you truly understand math, the less you’ll feel like you need to memorize. You’ll be far better off. $\endgroup$ – Clayton Sep 22 at 2:49
  • $\begingroup$ Thank you! Now I have more motivation to push through my mathematic studies. In your opinion do you think that math tests are a good evaluation of one's math skill? Theoretically, anyone could solve a quadratic with the quadratic formula but that's not what we're really looking for, are we? $\endgroup$ – Padarscha Sep 22 at 3:08
  • $\begingroup$ (1/2) I don't think you can group all maths tests into one category. There are certainly maths tests written by good professors, and maths tests written by lazy professors. The good professors try to encourage a solid understanding of concepts and test you on a wide variety of applications to make sure you do really understand the material, not just the formulas. $\endgroup$ – Toby Mak Sep 22 at 3:13
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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!


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?").
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