How do I think more mathematically?

  • Context: Mathematica 
  • Thread starter Thread starter STS816
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Discussion Overview

The discussion centers around strategies for developing mathematical thinking skills, particularly for high school students preparing for standardized tests like the ACT. Participants share their experiences and suggest methods to enhance problem-solving abilities beyond rote memorization of formulas.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant emphasizes the importance of practicing problem-solving rather than just applying formulas, suggesting that this approach is beneficial for standardized test preparation.
  • Another participant shares their personal experience of improving their ACT score through consistent practice and suggests seeking tutoring if difficulties persist.
  • A recurring suggestion is to focus on understanding algebra deeply, as it is foundational for deriving formulas and thinking mathematically.
  • Some participants advocate against memorization, recommending that students strive to understand the meaning behind mathematical statements and proofs instead.
  • One participant mentions engaging with problems outside of textbook material to enhance mathematical thinking, providing examples of problems that encourage deeper understanding.
  • A reference is made to a strategy by George Polya for tackling unknown problems, indicating that there are structured approaches to problem-solving that can be beneficial.
  • Another participant notes their current study of exponential modeling in precalculus as a positive experience that contributes to their understanding of mathematical concepts.

Areas of Agreement / Disagreement

Participants generally agree on the importance of understanding concepts over memorization, but there are varying opinions on the best methods to achieve mathematical thinking. No consensus is reached on a single approach, as multiple strategies are proposed.

Contextual Notes

Some suggestions depend on individual learning styles and may not apply universally. The discussion includes various levels of mathematical understanding and personal experiences, which may influence the effectiveness of the proposed strategies.

Who May Find This Useful

High school students preparing for standardized tests, educators looking for teaching strategies, and anyone interested in improving their mathematical thinking skills.

STS816
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I'm trying to figure out how to think more in terms of mathematics rather than going to straight to some formula or equation. I'm a junior in high school and I have a 97 in Pre Calculus so I'm not really having trouble with formulas and equations but I can't really seem to figure out problems that I don't know a specific formula for. I took the ACT today and I think this may have really hurt me on the math part.

Any tips on how I can become more mathematically inclined? Thanks.
 
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Just do lots and lots of problems (ones about problem-solving, not about plugging things into formulas) -- that's how one prepares for things like the SAT or ACT as well.
 
i can relate to you i have a high grade in pre cal to and wen i first took the act i had a 19 i just kept on practicing and now I am at a 30 around. its just practicing and timing, try getting a act or sat tutor if you are having difficulties its worth it in thne long run
 
STS816 said:
I'm trying to figure out how to think more in terms of mathematics rather than going to straight to some formula or equation. I'm a junior in high school and I have a 97 in Pre Calculus so I'm not really having trouble with formulas and equations but I can't really seem to figure out problems that I don't know a specific formula for. I took the ACT today and I think this may have really hurt me on the math part.

Any tips on how I can become more mathematically inclined? Thanks.

In one word: ALGEBRA.

Also study a laboratory science in which you will apply the introductory and intermediate levels of Algebra. If your intermediate course on Algebra includes some exercises on modeling, this can be very helpful in learning to think mathematically.

Back to the answer, "ALGEBRA"; this is the coursework in which you learn to derive formulas for yourself.
 
Another tip - Never set out to memorize a formula or an equation you see. Try to understand what the statement really means, explain it in simpler terms to yourself, know how to prove the statement. I often forget many identities but I remember how to prove them, I quickly prove them on the spot and I get them again. And don't memorize the proof either - just read a proof, several is possible, pick the one you find that makes most sense to you, it you understand what the proof did to achieve that result, you will remember it =]
 
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Memorization will ultimately fail you, and the only suggestion I can give is to try and engage yourself outside of your textbook math. You KNOW when you understand something and when you're just trying to memorize because you're in panic mode. I have developed over the years to reject not being able to understand something, and it drives me nuts when I can't because my memory sucks! Just an example of questions you might come across that I think help out mathematical thinking if you do enough of: Prove n^3-n is divisible by 3 (or the more difficult prove n^5-n is divisible by 5). I won't give you the answer, but there are all sorts of random problems out there.

I get some minimal practice by answering questions on this website! You never know what kind of weird questions people will ask that are thought provoking.
 
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symbolipoint said:
In one word: ALGEBRA.

Also study a laboratory science in which you will apply the introductory and intermediate levels of Algebra. If your intermediate course on Algebra includes some exercises on modeling, this can be very helpful in learning to think mathematically.

Back to the answer, "ALGEBRA"; this is the coursework in which you learn to derive formulas for yourself.

I'm doing exponential modeling in pre cal right now and surpising enough, I think I may really understand it.

And thanks Edgardo for that link.
 

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