I absolutely agree that students have to know how they will be graded, and the reasoning for doing it that way. What I think may help to resolve the tension you and I have over this is the implementation of the new Common Core State Standards, which demand far more written analysis and far less multiple choice testing. In both of our hypothetical cases here, this will avert the problem of all answers being potentially correct, since the student must mount a convincing (and accurate) rationale for his/her answer.

Disclaimer: The post was getting entirely too long, causing me to cut all of the background except for MsP's latest post.

I'm not seeing that in CCSSI. In fact, there are NO standards listed for any subject other than English Language Arts and Mathematics. Beyond the ability to read texts of the complexity found in science and history courses, there are no science and no social studies standards. The cynic in me wonders why. After all, it's not just in ELA and Mathematics where the "comparisons" of US students to the rest of the world fall short.

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From what I can tell in comparing sets of state standards over the last 20 years to the CCSSI, all they've done in mathematics is rearrange the deck chairs. Most states already had standards in place (some adopted in the first five years of the current century) that were nearly identical to the CCSS. They've spent a lot of time and money doing things that most states had already done. I'm not impressed, and while I agree that requiring students to justify answers would improve things dramatically, I don't think the implementation of CCSS will be the saving grace here.

Color me cynical if you like, but I think we spend more time talking about standards than actually doing anything with them. We've been working on mathematics standards for 25 years or more (nearly 50, if you want to start with the "new math" movement), and in the last 10-to-15 years, there hasn't been a lot of movement. It's time to stop reinventing the standards and to start actually training teachers to the standards and letting them teach their students to the standards. That means that it's also time for textbook publishers to actually write textbooks to the standards rather than repackaging the 1950s arithmetic and claiming that the book addresses the latest standards. I haven't seen a single text for K-5 mathematics that is actually, truly, completely aligned with the standards since I started teaching at the college level a decade ago, and implementation won't truly happen until we get one (but preferably, more).

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