Continuing the line of thought in my last entry, what happens when a list is sorted alphabetically or numerically? If a list such as this:
'4, 6, 2, 5, 9, 3'
is sorted into this:
'2, 3, 4, 5, 6, 9'
What happens when I do this? While I'm sorting, maybe I'm merely remembering that 2 goes before 3, 3 goes before 4, etc. I understand it more in terms of left-to-right and before/after than in terms of quantity. Maybe several things are going on here: I visually divide the list into distinct symbols (i.e., I see one list comprising six numerals); I remember that 3 always goes to the left of 4; and, either using the keyboard or using my imagination, I place 3 to the left of 4, and so on.
On the surface, it seems this thought process has little to do with quantities and the nature of numbers than it does with remembering an ordering of symbols. This is more clear when comparing two large numbers, such as the following:
121,432,131,272,983 and 121,432,131,282,983
In this case, I don't understand the magnitude of the numbers in question, I simply notice that 8 is greater than 7 and so I conclude that the first number is larger. When I say "greater than", I'm not sure if I'm simply in the habit of seeing the numeral 8 after the numeral 7, or if I somehow quickly see that 8 is quantitatively larger than 7. So, on second thought, maybe the way we "see" numerals does have something to do with their quantitative value, at least in some cases. But if I had to decide on the question now, I'd side with the notion from Meno, that it's a matter of recollection.
Wednesday, March 4, 2009
Monday, March 2, 2009
Writer's block. I was going to write in this blog every day, but this is more difficult than I thought it would be.
I was thinking about Meno, and the notion that learning is really recollection. That's my understanding of what the dialog says, anyway. I was thinking of how this might apply to an ordinary problem, such as ordering the following list:
x, f, i, g, m, h, z, t
If you're asked to arrange the elements in order, I suppose the only thing that's really happening is that you remember that f is before g, g is before h, etc.
Consider a list of numbers:
4, 6, 2, 5, 9, 3
As I'm ordering the list, do I understand that 2 is less than 3, or do I simply remember to place 2 before 3, because I've seen that ordering so many times, and it's merely something I remember, like a habit?
I was thinking about Meno, and the notion that learning is really recollection. That's my understanding of what the dialog says, anyway. I was thinking of how this might apply to an ordinary problem, such as ordering the following list:
x, f, i, g, m, h, z, t
If you're asked to arrange the elements in order, I suppose the only thing that's really happening is that you remember that f is before g, g is before h, etc.
Consider a list of numbers:
4, 6, 2, 5, 9, 3
As I'm ordering the list, do I understand that 2 is less than 3, or do I simply remember to place 2 before 3, because I've seen that ordering so many times, and it's merely something I remember, like a habit?
Saturday, February 28, 2009
I'll talk a little more about Meno now. It's been a long time since I read it in its entirety, so take what I have to say at this point with a grain of salt. If I'm off the mark, please correct me in the comments section. My understanding is that Socrates is saying that when we figure something out, when we discover a solution through reasoning our way to an answer, what we're doing is remembering or recollecting. I suppose that's his answer to the paradoxical question I mentioned yesterday, how can we find a solution if we don't know what to look for?
Socrates demonstrates the process of discovery through recollection by questioning a servant, a child who has not had any formal training in mathematics. Through a series of questions, the boy is able to figure out how to calculate the area of a square.
Since there is a lot of information and guidance in Socrates' questions, is this really a matter of recollection? At first, it seems that Socrates is teaching the boy by providing information in the questions. But it seems that the servant already has knowledge of the fundamental concepts. The first questions are, "Tell me now, boy, you know that a square figure is like this?" and "A square then is a figure in which all four sides are equal?" The boy answers yes in each case, so he already has knowledge of what a square is. Since a square (i.e., a two-dimensional object with perfectly straight and equal sides) is never seen in nature, how can he know what it is? We could say that someone taught him this, but then the question would be, how could he know what a perfectly straight, abstract line is?
As the series of questions demonstrates, the child not only knows what a square is, he knows enough to learn how to manipulate it. My guess at this point is that the child does learn something through Socrates' instructive guidance; he is learning a procedure for manipulating the square and discovering its properties. But maybe Socrates is saying that the properties discovered, as opposed to the procedure for discovering them, are recollected, not learned through instruction.
The child understands from the get-go that a square is divided into four parts when two lines are drawn through the middle, as shown in the diagram of the dialog. Where does this understanding come from? I have a hard time convincing myself that it came from Socrates. Socrates draws a figure, and the child naturally sees that one square is divided into four. He naturally sees that a figure can be decomposed into smaller, identical pieces. It's as if we are already born with knowledge of concepts such as one-as-many and division.
Socrates demonstrates the process of discovery through recollection by questioning a servant, a child who has not had any formal training in mathematics. Through a series of questions, the boy is able to figure out how to calculate the area of a square.
Since there is a lot of information and guidance in Socrates' questions, is this really a matter of recollection? At first, it seems that Socrates is teaching the boy by providing information in the questions. But it seems that the servant already has knowledge of the fundamental concepts. The first questions are, "Tell me now, boy, you know that a square figure is like this?" and "A square then is a figure in which all four sides are equal?" The boy answers yes in each case, so he already has knowledge of what a square is. Since a square (i.e., a two-dimensional object with perfectly straight and equal sides) is never seen in nature, how can he know what it is? We could say that someone taught him this, but then the question would be, how could he know what a perfectly straight, abstract line is?
As the series of questions demonstrates, the child not only knows what a square is, he knows enough to learn how to manipulate it. My guess at this point is that the child does learn something through Socrates' instructive guidance; he is learning a procedure for manipulating the square and discovering its properties. But maybe Socrates is saying that the properties discovered, as opposed to the procedure for discovering them, are recollected, not learned through instruction.
The child understands from the get-go that a square is divided into four parts when two lines are drawn through the middle, as shown in the diagram of the dialog. Where does this understanding come from? I have a hard time convincing myself that it came from Socrates. Socrates draws a figure, and the child naturally sees that one square is divided into four. He naturally sees that a figure can be decomposed into smaller, identical pieces. It's as if we are already born with knowledge of concepts such as one-as-many and division.
Friday, February 27, 2009
To rephrase the question I mentioned yesterday, what exactly happens when we have a problem before us, we think about it, and then arrive at a solution? Can we take snapshots of what happens in between the question and the answer? Maybe something like the think aloud protocol can help, and I'm interested in trying that out at some point.
On the surface, at least, what's happening is paradoxical. How is it that one moment I have only a vague, cloudy mental picture of something, and a few minutes later, without referring to information outside of myself, I have an answer? In Meno, an early dialog written around 380 B.C., the question is expressed like this:
One "cannot search for what he knows - since he knows it, there is no need to search - nor for what he does not know, for he does not know what to look for" [80e, translated by G.M.A. Grube]
The latter case is especially interesting. What happens when we think about a problem, and somehow, even though we may not be aware of what is happening, we see it more clearly?
It may be, in my case, that I don't see anything more clearly, I just see it differently. If I wonder how to proportionally reduce a rectangle, for example, and I arrive at a formula that works, I don't have to understand how it works. A more obvious example is a child who memorizes the multiplication table and learns algorithms such as long multiplication. He might become very adept at finding the answer to complicated equations without having an inkling of what he is doing or why he is doing it. How many students stop trying to learn a concept once they learn the procedure or algorithm for solving a problem? How many engineers, for that matter, learn just enough to get a job done? It's easy to confuse "I know how to solve it" with "I understand the answer."
On the surface, at least, what's happening is paradoxical. How is it that one moment I have only a vague, cloudy mental picture of something, and a few minutes later, without referring to information outside of myself, I have an answer? In Meno, an early dialog written around 380 B.C., the question is expressed like this:
One "cannot search for what he knows - since he knows it, there is no need to search - nor for what he does not know, for he does not know what to look for" [80e, translated by G.M.A. Grube]
The latter case is especially interesting. What happens when we think about a problem, and somehow, even though we may not be aware of what is happening, we see it more clearly?
It may be, in my case, that I don't see anything more clearly, I just see it differently. If I wonder how to proportionally reduce a rectangle, for example, and I arrive at a formula that works, I don't have to understand how it works. A more obvious example is a child who memorizes the multiplication table and learns algorithms such as long multiplication. He might become very adept at finding the answer to complicated equations without having an inkling of what he is doing or why he is doing it. How many students stop trying to learn a concept once they learn the procedure or algorithm for solving a problem? How many engineers, for that matter, learn just enough to get a job done? It's easy to confuse "I know how to solve it" with "I understand the answer."
Thursday, February 26, 2009
First post
Recently I started work on a project that required image processing for a traveling exhibit of historical illustrations. I was really looking forward to working on it, but when I actually began work, my enthusiasm faded, and I wasn't sure how to go about it. Also, I was surprised to find that I had to struggle a bit with a seemingly simple problem, re-sizing rectangles proportionally using Python. I thought about why abstract problem-solving can be a struggle, and now I'm interested in how I can make the problem-solving process more enjoyable.
The problems I'm concerned with aren't limited to those requiring an algorithm or formula, but abstract problems in general. The main question I have in mind is, "What happens when the solution to an abstract problem is discovered?" I hope that by pursuing this question, I'll make the process both more enjoyable and more effective. A related question is, "How can I apply what I know about abstract problem solving to software development?" The purpose of the latter question is similar, to make the process more interesting, more fun.
The problems I'm concerned with aren't limited to those requiring an algorithm or formula, but abstract problems in general. The main question I have in mind is, "What happens when the solution to an abstract problem is discovered?" I hope that by pursuing this question, I'll make the process both more enjoyable and more effective. A related question is, "How can I apply what I know about abstract problem solving to software development?" The purpose of the latter question is similar, to make the process more interesting, more fun.
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