Research on effectiveness of exercises in learning.
January 14, 2008 6:11 AM Subscribe
Does anyone know of research on the effectiveness of non-theoretical exercises (i.e.: finding a derivative of a simple polynomial when learning calculus, adding numbers when learning arithmetic, etc.)?
I'm sure there is some, but not being part of academia I wouldn't know the best way to go about finding it. Also, I'm lazy and metafilter is reliable. Lastly, I'm more interested in research about what sort of conditions make exercises effective and beyond what point they're not really improving learning that much.
Thanks!
I'm sure there is some, but not being part of academia I wouldn't know the best way to go about finding it. Also, I'm lazy and metafilter is reliable. Lastly, I'm more interested in research about what sort of conditions make exercises effective and beyond what point they're not really improving learning that much.
Thanks!
You might enjoy reading this PDF on teaching students to learn how to learn through Socratic exercises:
http://www.ibiblio.org/kuphaldt/socratic/doc/lett0045.pdf
posted by stungeye at 7:12 AM on January 14, 2008
http://www.ibiblio.org/kuphaldt/socratic/doc/lett0045.pdf
posted by stungeye at 7:12 AM on January 14, 2008
This not a particularly well-formed question, because it depends on what you mean by effectiveness. If you want to be able to differentiate polynomials (or, say, elementary functions) fluently then drill exercises are very effective at achieving this, and no amount of theory will substitute. On the other hand, they'll do nothing to prepare you to answer a question like "What is the derivative of a function?" Both parts are important, and mutually reinforce one another, and to be really successful, a student generally can't afford to ignore either. It's very hard to tackle even a moderately sophisticated multi-step problem if you can't treat elementary mechanical steps fluently. It's also hard to remember concepts, definitions, and theorem unless you have a context or framework for them that includes a generous amount of examples.
Another typical calculus "pit of drills" is techniques of integration. If you want to be able to antidifferentiate, there's no absolutely no substitute for doing lots of them. And while it is, in a sense, a fairly limited domain, both the design and the solution of integration problems call for creativity, and - importantly! - involve problems where there are many potential techniques available and you have to begin honing your personal sense for when and how a particular technique (or theorem) is likely to help make forward progress. That adds a real level of depth to the understanding of theory, and if you're attentive, it also tends to help you see what little variations and extensions are possible on the techniques.
It is very, very rare that a single course in a subject can provide well on both practical and theoretical grounds. Linear algebra is a subject where you see a remarkable split, as people tend to teach the subject with two rather different styles. There's the theory-oriented style (they like Halmos's book), characterized by students who finish the course "knowing everything about the determinant" (for example), but would fail utterly at computing one if you gave them a four-by-four matrix. Then there's the computational-style course whose successful students can carry out G-J Elimination, compute inverses, find a kernel, write a rotation matrix - but have no idea, for example, what an eigenvalue really is, besides a thing you find by doing the find-an-eigenvalue thing. Almost everyone really needs to go through linear algebra twice (or more) to get a balanced picture that addresses what, why, and how equally well. The same is true of calculus and arithmetic and algebra and so on, of course.
posted by Wolfdog at 7:55 AM on January 14, 2008
Another typical calculus "pit of drills" is techniques of integration. If you want to be able to antidifferentiate, there's no absolutely no substitute for doing lots of them. And while it is, in a sense, a fairly limited domain, both the design and the solution of integration problems call for creativity, and - importantly! - involve problems where there are many potential techniques available and you have to begin honing your personal sense for when and how a particular technique (or theorem) is likely to help make forward progress. That adds a real level of depth to the understanding of theory, and if you're attentive, it also tends to help you see what little variations and extensions are possible on the techniques.
It is very, very rare that a single course in a subject can provide well on both practical and theoretical grounds. Linear algebra is a subject where you see a remarkable split, as people tend to teach the subject with two rather different styles. There's the theory-oriented style (they like Halmos's book), characterized by students who finish the course "knowing everything about the determinant" (for example), but would fail utterly at computing one if you gave them a four-by-four matrix. Then there's the computational-style course whose successful students can carry out G-J Elimination, compute inverses, find a kernel, write a rotation matrix - but have no idea, for example, what an eigenvalue really is, besides a thing you find by doing the find-an-eigenvalue thing. Almost everyone really needs to go through linear algebra twice (or more) to get a balanced picture that addresses what, why, and how equally well. The same is true of calculus and arithmetic and algebra and so on, of course.
posted by Wolfdog at 7:55 AM on January 14, 2008
I am RA to a maths academic and I've been entering the articles she's saved into Endnote and I can tell you, there are thousands of academic discussions on drill and practice (thank you Wolfdog for the terms) and related issues. There's articles on cognitive processes, metacognition and the psychology of teaching and learning maths. It's huge.
If you really care, I would suggest you start with google scholar, access to a good electronic library and the search terms drill, practice, mathematics and then check out the references in those articles and so on.
posted by b33j at 12:33 PM on January 14, 2008
If you really care, I would suggest you start with google scholar, access to a good electronic library and the search terms drill, practice, mathematics and then check out the references in those articles and so on.
posted by b33j at 12:33 PM on January 14, 2008
In mathematics, it is possible to learn a great deal without knowing anything. Proving stuff is good, but without being adept at computation, you will never be able to figure out what to prove, or why any particular thing should be proven. Any decent math book will have computational problems, even if the text is 100% "Let P be a ...", and if you just learn the text, and omit the exercises, you will have learned nothing.
And practice alone is worthwhile. Here is GB Shaw on it, from Cæsar and Cleopatra:
CLEOPATRA (to the old musician).
I want to learn to play the harp with my own hands. Caesar loves music. Can you teach me?
MUSICIAN
Assuredly I and no one else can teach the Queen. Have I not discovered the lost method of the ancient Egyptians, who could make a pyramid tremble by touching a bass string? All the other teachers are quacks: I have exposed them repeatedly.
CLEOPATRA
Good: you shall teach me. How long will it take?
MUSICIAN
Not very long: only four years. Your Majesty must first become proficient in the philosophy of Pythagoras.
CLEOPATRA
Has she (indicating the slave) become proficient in the philosophy of Pythagoras?
MUSICIAN
Oh, she is but a slave. She learns as a dog learns.
CLEOPATRA
Well, then, I will learn as a dog learns; for she plays better than you. You shall give me a lesson every day for a fortnight. (The musician hastily scrambles to his feet and bows profoundly.) After that, whenever I strike a false note you shall be flogged; and if I strike so many that there is not time to flog you, you shall be thrown into the Nile to feed the crocodiles. Give the girl a piece of gold; and send them away.
posted by hexatron at 1:06 PM on January 14, 2008
And practice alone is worthwhile. Here is GB Shaw on it, from Cæsar and Cleopatra:
CLEOPATRA (to the old musician).
I want to learn to play the harp with my own hands. Caesar loves music. Can you teach me?
MUSICIAN
Assuredly I and no one else can teach the Queen. Have I not discovered the lost method of the ancient Egyptians, who could make a pyramid tremble by touching a bass string? All the other teachers are quacks: I have exposed them repeatedly.
CLEOPATRA
Good: you shall teach me. How long will it take?
MUSICIAN
Not very long: only four years. Your Majesty must first become proficient in the philosophy of Pythagoras.
CLEOPATRA
Has she (indicating the slave) become proficient in the philosophy of Pythagoras?
MUSICIAN
Oh, she is but a slave. She learns as a dog learns.
CLEOPATRA
Well, then, I will learn as a dog learns; for she plays better than you. You shall give me a lesson every day for a fortnight. (The musician hastily scrambles to his feet and bows profoundly.) After that, whenever I strike a false note you shall be flogged; and if I strike so many that there is not time to flog you, you shall be thrown into the Nile to feed the crocodiles. Give the girl a piece of gold; and send them away.
posted by hexatron at 1:06 PM on January 14, 2008
This thread is closed to new comments.
posted by mzurer at 7:03 AM on January 14, 2008