Sunday, July 1, 2012

Two ways to ensure learning

When students get the right answer, how can you be sure that they did so via the skill that they're supposed to be learning? This is particularly problematic in automated learning situations, where there is no third party monitering the student. When I click on the correct picture in a Rosetta Stone Spanish exercise, how can one be sure that I didn't get the right answer simply because I recognized a single key word in a sentence, as opposed to having correctly parsed out the syntactic structure being taught?

When it comes to language learning, the best way to ensure that students are exercising the intended skill is to make displaying it an inherent part of getting the right answer. If you're teaching grammar, for example, answering correctly should involve actually producing a sentence with the grammatical structure being taught, not just clicking on a picture.

Ensuring that students are applying the intended skill is one of the justifications for today's obsession with having them explain their answers to math problems. But that's the easy way out, and it frustrates students and decelerates their progress. An alternative would be construct math problems whose solutions are unlikely to be found by any means other than by using the skill in question: arithmetic problems that involve more than single digit or "friendly number" solutions; algebra problems that don't lend themselves to arithmetical, guess & check solutions. Then there are problems complicated enough that there's actual work to show; work that, if written out systematically, helps the students at least as much as the teacher.

Constructing such problems isn't easy--they are studiously avoided by the various Reform Math curricula, which consistently place the burden on the students to explain their answers. But, when it comes to ensuring that the relevant skills are being exercised, shouldn't the burden be on the deep-pocketed textbook companies and their teams of professionals rather than on the students themselves?


Daniel Ethier said...

I think you hit the nail on the head here. I've been saying this about math for years.

Math contests, like the AMC-10, are a good place to look for these kinds of harder problems that you can only do if you understand.

There are also plenty of books of contest problems that have some good ones, although it can take some work to find them unless they have a good index.

gasstationwithoutpumps said...

The Art of Problem Solving books ( are an excellent source for challenging problems. Note that all the old AMC tests are also available on the web.

Anonymous said...

When I was in middle school I participated in a math olympiad. When the results came out I found out that I had done correctly 2 problems and had got 1/2 credit for the third one. When I saw the teacher I expected to get criticized for the one problem I hadn't done correctly. Instead he was beaming with pride, I had won first place. And thus I learned that victors do not get judged.

When I do math with my daughter I never ask her how she did it if she gives the correct answer. Apart from the fact that it would be very boring for her to explain it to me, I think that even if it were obvious she was guessing, being able to guess is a good skill to have.

Instead I make sure that problems I give to her are difficult enough that she gets a few of them wrong. Then we can discuss in detail how she did it and I can correct any errors in her understanding.

Anonymous said...

nice posting.. thanks for sharing.