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I'm currently doing Calculus I (one variable and some max- min problems with two variables as well), and I can follow most proofs and understand most of all chapters in the book. It seems tough that when solving problems you really need to be creative, and it is this creativity that I lack at the moment. For instance, limits is of course a good example where it is not enough with just knowing the theory but you also need to be creative to find a solution. It seems like I have to manipulate expressions all the time before actually solving problems (now I'm speaking of calculation problems, not "show that.." problems).

Is it just me or is there some "creative structure" I can follow when solving Calculus problems? There are so many different things to have in mind, and it feels like this course is almost going more on to memorizing formulas rather than actually understand them, which is frustrating.

It just seems that knowing the theory really good is not just enough?

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up vote 27 down vote accepted

Many people get through Calculus by memorizing formulae. But you'll learn best by combining "conceptual understanding" with "procedural knowledge".

To answer your question regarding how to develop the "problem-solving" creativity needed to manipulate expressions into forms you can apply theory:

Answer: Practice! AND Effort (Perseverance)!, AND Time (Patience)!

  • Practice: Math is not a spectator sport! The more problems you encounter, the more strategies and techniques you'll encounter, and interacting with both problems and their solutions will build your "repertoire": some tools you'll be able to use when encountering similar problems.

  • Perseverance: Keep at it! What might seem like "tricks" at first, or what might seem as "creative" now, will become "second nature" to you in no time at all, if you acquire "working knowledge" of how to use the "tools" you acquire!

  • Patience: Already addressed, in part. Proficiency, creativity, and mastery can be developed and nurtured with effort, practice, and time. No need to feel intimidated if you don't immediately "get" it. The creativity you speak of is a reflection of the collaborative work of mathematicians over time, each learning from one another...No one "knows it all" from the "get-go."

So, in short: You'll be on your way if you commit to the "three P's":

Practice, perseverance, and patience.

EDIT - One point of observation: The mere fact that you are questioning your strategies, wondering about creativity, thinking about how to merge conceptual understanding with procedural efficacy when attacking problems: these are all indicators that you are engaging in "meta-cognitive" scrutiny. That is, you're thinking about how you think mathematically, questioning how to best learn, and scrutinizing your current "plan of attack" with the aim to develop greater procedural flexibility in applying what you're learning accurately and creatively. These are all good things. Self-awareness and self-examination are key components to effective learning, and they indicate your recognition that you are an active player in your own education.

One book I highly recommend to anyone/everyone serious about mathematics is the book authored by Mason, Burton, and Stacey' Thinking Mathematically. When you're not under the grind with homework and/or exams, or can otherwise find the time, you might want to "have a read."


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I like this answer, I struggle with being "creative" when it comes to solving math problems as well. I actually thought that I wasn't good at math just because I wasn't as creative as my math peers, but an answer like this is reassuring that it just takes time. Thanks @amWhy. – TheHopefulActuary Dec 19 '12 at 15:07
One of the points about practice is that you need to be able to see how to attack a problem. Practicing lots of "Integration by parts" examples will allow you to do those questions, but you'll need to master each bit and then do plenty of more complex ones so you can apply it - this is the bit that isn't just knowing the formula - recognising the substitution to perform and the order to do it. – Philip Whitehouse Dec 19 '12 at 21:27

Quick answer: you are right, theory is definitely not enough. That's why many people hate mathematics, because you must both study the theory and develop some practice with problem-solving. In other words, you have to train a lot, after studying theorems and proofs. Knowing the theory is a necessary condition, but not a sufficient one.

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Apart from the other answers, some points:

(1) Try to do the same exercise in many different ways!

(2) Draw figures. This is extremely important!

(3) If there are exercises with hints, try first to do it without reading the hints. If that does not work, after solving it with the hints, apply point (1) above.

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+1 'try to do the same exercise in many different ways!' This is a great way to build up a better understanding/instinct as to what works well with which types of problems. – Mr.Mindor Dec 19 '12 at 19:18
+1 for drawing - most of calculus isn't too abstract, so take advantage of that! – Sean Allred Dec 19 '12 at 21:26
One bok you should read to get many hints on how to apply my principles: G Polya: "How to solve it". Buy it , read it , PRACTICE it! – kjetil b halvorsen Dec 19 '12 at 23:04

I wholeheartedly agree with what has been said above. My calculus professor in college said that Cal 1 is 10% calculus, and 90% algebra. It always felt like we applied a little calculus somewhere and a whole lot of algebra to solve the problem, so my tip for Cal 1 would be to make sure your algebra skills are tip top so you can easily spot trig substitutions, polynomial factors, etc., since as you say you have to manipulate the expression (algebra) before you actually solve it (most likely with some quick calculus theorem).

I would make the same recommendation for Cal 2 depending upon your curriculum, only substitute algebra for derivatives: being a derivatives ninja should make Cal 2 much easier.

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It helps to know that calculus is literally only a tool you use on top of everything else you've learned. A sharp wit and a keen eye are your best friends throughout the whole of mathematics. – Sean Allred Dec 19 '12 at 21:28

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