On the Importance of Hard Tests

by rrusczyk, Jan 2, 2011, 7:25 PM

AoPSer sophia sent me this article on the importance of testing, and most specifically the importance of difficult tests, to the learning process.

It's generally pretty unpopular in most education circles to advocate the value of testing, but I think this article does get at how middle and high school math contests prepared me so well for all my classes in college (and not just math classes). Well-designed tests, with difficult questions and near-immediate reflection on your performance, are outstanding at hammering home lessons. Imagine trying to learn how to ride a bike or play a musical instrument without testing (what others might call "practice"). Simply impossible.

education

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It is funny how the educators first build various "groundbreaking theories" and then, when everybody is brainwashed enough, return to old common sense and put it forward as a next discovery. Let's wait until they come forward with a bold new idea that teaching calculus once with all rigor necessary is more efficient than teaching it three times waving hands each time. Anyway I haven't thought I'd live to hear the words

Quote:

Fear and failure are good motivators

said openly by an US educator (even as a quote from a student).

by fedja, Jan 2, 2011, 11:39 PM

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I'm quite sympathetic to the conclusions of this article. I'm wondering if it contradicts an article I read recently in the Notices of the AMS:
http://www.ams.org/notices/201010/rtx101001303p.pdf.
The Notices article seems to say (near the end) that studying worked examples before solving problems is important. (I added the word "before", but that's my recollection of what the references cited in the Notices article say.) The article you linked to says (about three-fourths of the way down) that attempting questions before studying the answers is important. Now I'm not sure what to think.

by Ravi B, Jan 4, 2011, 2:17 AM

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I'm guessing a lot of it depends on the topic, and on what skills we're actually trying to impart to kids. "Find a solution to a new problem" is, to me, among the most important skills, if not the single most important skill, that we can hope to give our kids. It seems almost tautological to say that the "try the problem first" approach is better for this than the "do worked examples, then repeat" approach. But I could see the latter approach working better for some specific skills.

by rrusczyk, Jan 4, 2011, 6:51 PM

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@Ravi I guess neither extreme is a good idea that should always be a preferred way. IMHO, it is good to let the students try the new problem without any hints when there is a decent chance they'll be able to do it based on what they had before. Also, that is something to do at home, not in the classroom. The process of absorbing new techniques from examples presented by a teacher can be synchronized but the process of figuring things out is different and differently paced for everybody. I like most what is the unwritten policy for homework help on AoPS: try, show where you are stuck, get some hints, try again, etc. It is neither killing their initiative making them finite automata that are capable of just invoking standard algorithms, nor making them break their heads against the walls that are just too strong for them at the moment. There is no need to believe anything, by the way: just give your students assignments of both kinds and see which work and which do not.

by fedja, Jan 5, 2011, 9:41 PM