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Friday, October 11, 2013

The Meno Problem

If you've ever read Plato's Meno, you probably recall the conversation in which Socrates tries to prove to Meno that all knowledge is recollection by helping Meno's slave boy recall some geometry. Socrates does this in his typical style, through a series of questions:

Soc. Mark now the farther development. I shall only ask him, and not teach him, and he shall share the enquiry with me: and do you watch and see if you find me telling or explaining anything to him, instead of eliciting his opinion. Tell me, boy, is not this a square of four feet which I have drawn?
Boy. Yes.
Soc. And now I add another square equal to the former one?
Boy. Yes.
Soc. And a third, which is equal to either of them?
Boy. Yes.
Soc. Suppose that we fill up the vacant corner?
Boy. Very good.
Soc. Here, then, there are four equal spaces?
Boy. Yes.
Soc. And how many times larger is this space than this other?
Boy. Four times.
Soc. But it ought to have been twice only, as you will remember. 

Boy. True. 
Soc. And does not this line, reaching from corner to corner, bisect each of these spaces?
Boy. Yes.
Soc. And are there not here four equal lines which contain this space?
Boy. There are.
Soc. Look and see how much this space is.
Boy. I do not understand.
Soc. Has not each interior line cut off half of the four spaces?
Boy. Yes.
Soc. And how many spaces are there in this section?
Boy. Four.
Soc. And how many in this?
Boy. Two.
Soc. And four is how many times two?
Boy. Twice.
Soc. And this space is of how many feet?
Boy. Of eight feet.
Soc. And from what line do you get this figure?
Boy. From this.
Soc. That is, from the line which extends from corner to corner of the figure of four feet?
Boy. Yes.
Soc. And that is the line which the learned call the diagonal. And if this is the proper name, then you, Meno's slave, are prepared to affirm that the double space is the square of the diagonal?
Boy. Certainly, Socrates.
Soc. What do you say of him, Meno? Were not all these answers given out of his own head?
Men. Yes, they were all his own. 


I often try to teach Socratically, using questions to guide students into answering their own questions. Recently, though, a comment from a student got me thinking. She complained that when I'm helping her, she feels like she understands the concept perfectly, but when she goes home and starts here homework, she has no clue what to do. I know she can't be the only one who feels this way. So does this question and response style of teaching actually work?

I have to admit I doubt Socrates' claim that all of his pupil's answers were "given out of his own head"--some of these answers are so obviously fed to the boy that it makes me wonder if Plato was being serious when he wrote this. I doubt even more that the boy took away a lasting understanding of the relationship between linear dimensions and area. My question, then, is: how do good teachers use questions to guide student thinking in a way that both gives students structure and enables them to become independent of a teacher's guidance? Any ideas out there?

5 comments:

  1. Ryan, your thought line here is a good one, involving good questions. I certainly don't pretend a systematic or comprehensive response. That could be lengthy, and I doubt I have the innate ability or pedagogical experience. With these qualifications, permit me a few simple thoughts, thoughts which probably only echo ones you have already had. -- (1) Perhaps not all learning can be conveyed through a Socratic or Socratic-like method, but I think you are right to attempt it as much as possible. Engage the other person in a process which requires more rather than less effort on his or her part; enable a person to dig his or her own garden as much as possible, with guidance based on knowledge and experience the person does not have. (2) I have found from personal experience that learning is a process over time, involving repetition and fermentation. However good and deep my initial exposure to something was, it always took a while to absorb what I had learned; that is, letting it sift through my conscious and seven subconscious mind to sort into and add to or change categories of thought. In keeping with this, going over it again, especially from different angles (e.g., a different book on the topic; a second lecture; a third attempt at problem-solving) contributed to this deepening and extending of my knowledge. A student will know it better over time and through repetition; at least many or most will, especially if motivated at all to learn. (3) There is almost always a deepening and extending of knowledge by making it one's own; that is, by "owning" it, not simply receiving it. One "owns" it by absorption over time and repetition. One also "owns" it by passing it on to another; that is, by explaining or teaching it to another. Perhaps this last part could be incorporated somehow into your classroom experience? For example, one student to another, or a student to a parent or sibling? Some other situation? -- Just some thoughts. Keep up the good work!

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  2. I had a conversation along these lines on Thursday. One of my Calc BC students is a star in class conversations but he is meandering and aimless in his work on timed assessments. I spoke to him about his ability to follow my questioning prompts and I urged him to make note of my questions. I told him that his job is to start asking himself the same kinds of questions that I ask of him in class. I know it is easier said than done, but I also know that I was at least giving him a beginning step.

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  3. Great thoughts, folks! Thanks for the help!

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  4. While I get what Socrates was trying to do, I don't think he was actually asking questions. If anything, he was making statements with inflection designed to illicit a specific response. I do believe strongly in the Socratic method and also try my best to ask questions instead of give information.

    The amazing Fawn Nguyen put it perfectly: The more I talk, the less they learn.

    I think a great way to do this is to try to ask questions that do not allow for a single word response as often as possible.

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  5. This is why conversations with math teachers is so meaningful. I completely agreed with the beginning premise that sometimes I feel that I use the Socratic method and the kids go away having said the "right" answers, but unable to do the work on their own. Then Justin's comment cleared up what I was doing wrong, I am asking too many single word response comments where I lead them to the answer. I will have to watch that tomorrow as I teach.

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