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?
Best Answer
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."
Regards.