Not all knowledege can be thought of being stored into separate declarative and procedural types of memory. For example, I think that memory type distinction is very blurry in mathematics, if it exists at all. To understand mathematics, you have to follow a procedure in your head – an algorithm. If you take a piece of paper and write some mathematical principles and their derivations down, that knowledge is still theoretical and declarative. Mathematics is an abstraction of concrete world scenarios. But I can still live in that abstract world, create more knowledge, understand it, and it can all happen in my head. Given the human knowledge of mathematics and the symbol system that has been created to write it down, is the knowledge of arithmetic just a « belief »?
Maybe I am misunderstanding your point. I guess you are focusing here on knowledge that requires a concrete medium in other to be transmitted and understood, for which you need to develop a skill that allows you to do it without a big conscious effort.
p.s.: I like the way you draw hands :)
I don’t know. I’ve been thinking about this, this chapter is more or less written since a long time and I’m not quite sure I still agree. i’m not very advanced in mathematics, but this reminds me of when I took this pre-calculus course on Coursera. I only listened to the the first chapter, which was a review of what you’re supposed to already know (of course I already learned it in high school but it was a looong time ago, so I don’t remember a thing). The teacher was explaining the priorities of equations, and how to multiply exponents, and such algebra things, while I was eating my spaghetti. I was looking and understanding (yeah of course! obvious, duh!) but when I tried to do the exercices by myself, I realised that even if I understood what the teacher was saying, I was not able to reproduce it by memory. (what was the rule again?….). As you say, I would follow a procedure in my head, but I have to have *processed* this procedure before….
Maybe I’m just dumb (that’s a possibility!), maybe I’m just absolutely unable to learn anything without actually doing it a couple of times. :P
Anyway, this distinction between procedural and declarative memory is way too simple to reflect what’s happening in our heads. It’s just a convenient theorical start point to explain things I observe.
You are definitely right in that mathematics requires practice! But I think the type of memory that is being trained coudl still be considered declarative. Although now I am confused, because I see most mathematics as procedural. Maybe the declarative part is an easier way to remember concepts? First you learn a bout them by hearing them. Then, you try them out by applying the concepts to a particular problem, and convince yourself that the concepts are true. Afterwards, you can use the concepts without having to remember why you were convinced they were true or useful.
I guess most of us need to find a concrete example of the abstract concepts in order to really understand them. But then, I don’t really know what it means to understand anything. If you are good at producing music of drawings, could you say that you understand what makes you good?
This reminds me of a problem in computer science, which is called the P=NP problem. In short (and probably a bit innacurately), the problem is to demostrate wheter if finding a solution a problem is as easy (or as hard) as verifying if an existing solution to the problem is correct (check some of the blog posts by Scott Aaronson). If this were true, then you would only need a appreciate a work of art in order to be able to reproduce, and to produce something that would be considered equally as good/beautiful/etc. Certainly, we require something more than just following instructions in order to become good at music/painting/drawing/math. Maybe practice gives us that something extra that we require. But could we still call it knowledge if we cannot really explain it?
Je suis desolé d’avoir écrit tout en anglais. Je sais que je dois pratiquer le Français plus souvent. Mais il me faudrait des millions d’heures pour écrire quelque chose. C’est vraiment difficile pour moi, le Français.
Your french is not yet in your procedural system. ;) Don’t worry about it.
I think the declarative part might be just the first step towards the procedural. This is where you use all those mnemonic tricks in order to remember the steps of something, until you’ve practiced it enough to do it without using the tricks. This is more or less what you’re saying, I guess.
I use the word « understand » (comprendre) to talk about how I get better in drawing, but the understanding is really really small; if in an epiphany I understand that a human jaw usually have this shape, it doesn’t mean that I understand how to draw a human arm. There are infinite things to understand in each domains, and some are more fondamental then others – but you cannot « understand » something completely.
I don’t really calculate my understanding of drawing by looking at how « good » it is, that’s not possible to me (good compared to what??). But I do feel that the number of parameters i’m aware of is increasing (like composition, rythm in the page, schematization, etc), and the older parameters i’m familiar with (like facial expressions, etc) are much easier to do than before.
Your P=NP problem suggest that there is a solution, and that we can become « good » at drawing. That notion of « good » is the most unclear thing. When you have a problem in computer science, maybe you can know whether it’s solved or not. But can you say of a music composition or a painting that it is « solved »? There is no problem in the first place!…
(j’aurais dû écrire mon message en français ;) )
voilà, juste un exemple entre x exemples
« Peindre non la chose mais l’effet qu’elle produit » (Mallarmé)
« Chez moi la chose et l’effet qu’elle produit ne font qu’un » (attribué à l’on ne sait pas trop qui)
Gualtieri a commencé en plein triomphe de l’abstrait, les galeristes 1950 haussaient les épaules lorsqu’il leur decouvrait ses toiles
Aujourd’hui il a un musée personnel du côté de Rimini
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