3 ms·
Not the intended target, but I'll throw in my two c from doing a lot of applied math. Math readily has two components. The first is a formulaic, formal compone
by tel 16y ago
Not the intended target, but I'll throw in my two c from doing a lot of applied math.
Math readily has two components. The first is a formulaic, formal component that can be readily overcome by rote. The second is the more freeform conceptual understanding that motivates and directs the first. I feel confident that if you ask anyone familiar with advanced math if they understand concept/theorem/tool X, they'll say yes if they know it in the second form and are confident that they can reconstruct the important parts of it in the first.
I think a lot of why people rebel against rote memorization then is that it, as a method, is very likely to prevent you from encountering the second side there. If you honestly use it to improve your fluency with the formal manipulations, it can be a great tool for learning more math. It's just easy to lose that honesty.
To really understand math, you need to recognize that it's a language you must both read and write. I suggest that if you do get strong benefits from rote memorization, then you should complimen t your reading by attempting to synthesize mathematical concepts you've not seen before. Read the claim of a theorem and then prove it yourself without knowing the answer. If you can honestly complete mathematical synthesis at that level as well, then rote memorization isn't hurting you in the least.
- whatwhat 16y agoThanks. That's a great response.
- jacobolus 16y agoTo elaborate on this, a working knowledge of some area of mathematics is not like a set of historical facts to be familiar with, or a list of fundamental particles and their properties, or a group of plays or novels to be quoted from, or a set of pigments and their interactions with brushes and paper, or even a code library’s API. Mathematics is, fundamentally, about model building. The study of mathematics is about learning how to make maps even more than it is about the specific territory being mapped. In my opinion the largest part of mathematical fluency is the constant willingness to test mathematical structures and ideas against each other and against new data, to figure out how parts work at their deepest levels and then to go back and try to see how each one fits with all those known before. What matters in understanding a mathematical concept is not whether you can repeat a witnessed proof step by step or write down a formula, but whether you have an intuitive grasp of the abstraction(s) in question, whether you can explain them to yourself (an ability to explain them to others also recommended), and whether you can apply them to new problems which arise. It is my belief that this kind of deep understanding and fluency can only be obtained by repeatedly interacting with these abstractions in a wide variety of problems and contexts, writing down the patterns and working through the proofs, questioning the axioms underlying them, asking how they generalize or how they apply to specific cases, and so on. Very little of this work can be done on flash cards, at least for me personally. Indeed, I believe it is precisely the teaching of mathematics as something which can be learned from flash cards which most impedes mathematical education and understanding. See http://www.maa.org/devlin/LockhartsLament.pdf http://www.maa.org/devlin/LockhartsLament.pdf
- whatwhat 16y agoThanks.