#1. A true mathematician is a pattern-seeker. The majority of math concerns patterns in numbers, but a skilled mathematician finds himself well equipped to deal in logic and philosophy as well - because it is not numbers, intrinsically, but patterns, that concern him. The better you are at noticing, conceptualizing, defining and explaining patterns, the better you are at mathematics.
#2. A true mathematician relishes challenge. Never work out on paper what you think you might be able to do in your head. Take ten seconds to think about it. If you aren’t sure, then check it by working it out (when you’re being graded, always work it out). Likewise, only use the calculator to save time! Never use it as a magic conch! If you can’t explain to yourself or someone else how the calculator got the answer it did, you are doing divinations, not mathematics!
#3. The first and most valuable weapon of the mathematician is his form – the legibility, completeness, and formality of his writing. A beginner mathematician’s proficiency in the discipline will be determined largely by his writing habits, not an absence or presence of natural pattern-recognition ability. Developing proper writing habits for each mathematical statement and operation is a lifetime endeavor – it is to mathematics, as adopting the proper stance and delivering the perfect strike is to martial arts. An effective personal notation is one that lessens the amount of thinking or writing you have to do, while still enabling you to come up with the correct solution every single time. There are several general recommendations that age and consensus have bestowed upon the modern mathematician –
3a. Invest in exactly one durable mechanical pencil of extremely high quality and grip comfort. If you don’t like mechanical pencils because the graphite breaks too often, get one that uses ultra-thick graphite. Both the graphite and the eraser (and it should definitely have one of those) should be replaceable. Keep resupply in the internal cartridge at all times. This tool will be your Shaolin spade, and with it you shall cleave your wisdom and solutions unto the universe. Having only one of it forces you to properly anticipate when you’ll need it, while motivating you not to ever lose it, which promotes overall organized behavior. Also have one big pink eraser with you for those times when you need to erase en masse that whole page you just wasted ten minutes on. Incidentally, never do math in pen, unless you have no other option.
3b. WRITE LIGHTLY and smoothly. Assume you’ll be erasing it in five seconds. Nurture an intense hatred of those highly visible eraser smudges smeared across part of the paper that obscure work written over them.
3c. Work in horizontal rows, giving each “statement” its own row, and each written operation its own row as well. If there isn’t enough room on the paper beneath the statement of the problem as it was presented to you, your mathematical training is being actively sabotaged by a cavalier or malevolent instructor. Get out your spare loose-leaf sheet or notebook and work out the problem where there is sufficient space. Give yourself as much room to write out each row as you think you’ll need. For example, if you have to rearrange the following equation to get X by itself:
3X + Y + Z = 20 Tackle the exact original statement where there is room
- Z - Z operation
3X + Y = 20 – Z statement
- Y - Y operation
3X = 20 – Z – Y statement
/3 /3 operation
X = (20 – Z – Y)/3 destination statement
Notice how the “equals” signs form a kind of backbone of rivets in a good notation. That’s because they are the backbone. You can’t stand up to fight without one, and neither can your equations. Bring the equals sign with you at every step!
3d. Legible handwriting should go without saying, but I’ll say it anyway. Ask a friend to read back to you what you’ve written, if there is any discrepancy between what they say and what you meant – either because your Z looked like a 2 or your 5 looked like an S or one of your numbers was just scribbled too quickly to be readable – this is an area you need to work on. Take your time if you have too, don’t rush! A true mathematician will see the beauty of what they are writing (given that they understand it) and so endeavor to render it in equal beauty. This equal beauty will promote less error-prone mathematics, and it will constitute a positive feedback loop between what you are doing and how you want it to appear. Bad handwriting will promote systematic error, which will in turn promote a lack of respect for the math, a negative feedback loop.
These are the four sacred principles of good form together – using a good pencil to write lightly and legibly, while working in rows.
#4. The second and most versatile weapon of the mathematician is his technique – how many known patterns in his repertoire he can effectively utilize, when, how, and why. This will be determined initially by the level of math course that you are in, yet as you grow in skill, the true mathematician’s spirit within you may awaken to the contemplation and investigation of patterns beyond the current curriculum. Learning to willfully summon and control this spirit at the times of your greatest need is one of the greatest boons of a mathematician’s training, for it is the spirit of independent learning, and the spirit of the historical enlightenment that brought man from the dark ages to the age of science.
The "big 8" essential techniques leading up to and into Calculus may be (very) briefly summarized as follows:
The "big 8" essential techniques leading up to and into Calculus may be (very) briefly summarized as follows:
Addition: A + B = C entails Subtraction: C - B = A
Multiplication: A(B) = C entails Division: C/B = A
Exponentiation: AB = C entails The Logarithm: LogA(C) = B
Derivation: (dy/dx) = f'(x) entails Integration: ∫ f'(x)dx = f(x)
Even if you are not fluent in the notation for these ideas, a certain fact warrants mentioning. Every single one of these operations is really just an abbreviation of some kind for . . . addition. If you're smart enough to add, you're smart enough to integrate. It's just a matter of:
A. Comprehending the pattern being referred to.
B. Becoming fluent in the notation used to represent it.
It also warrants mentioning that you should get used to using parentheses to denote multiplication, rather than the "dot" or "X" that some teachers employ for the concept in earlier level math courses. Why? Numerous reasons. For one, both the "dot" and the "X" represent entirely different and significantly more complicated operations in Calculus (the dot product and cross product, respectively). More immediately apparent for those in earlier math courses, "dots" are easily mistaken for decimal points and "X"'s are easily mistaken for instances of the variable x, if one's legibility is sloppier than it ought to be. Assuming that you're striving for a notation that is as unambiguous and accessible as it can possibly be, stick with parentheses.
#5. The third and most venerated weapon of the mathematician is his error sense – how quickly, accurately, and with what degree of poise and resignation he has learned to notice and correct his own errors, and consequently, the errors of others. If you've attempted a practice problem but have come up with an incorrect answer, you should probably do the following:
5a: Check that you have copied all parts of the problem down correctly.
5b: Do the problem again, checking all the "specifics" - making sure everything really adds or subtracts or multiplies or divides to what you said it does.
If those two steps haven't produced a correct answer, it means that your error is operational in nature. Somewhere along the line, you are performing an operation (doing something) that you just plain aren't allowed to do in general.
5c: Mentally review and justify to yourself why you believe you are allowed to take each step you've taken in getting to your answer.
Immediately after you've done these three things (if you still haven't gotten what the solution manual got) is the most productive point at which to ask for help, because you have eliminated the possibility of random error and still have the concepts and specifics of the problem fresh in your mind.
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