David Truss has a great post called Assessment & Rote Learning: Math Conundrums. I tried to leave a comment. I discovered I'm passionate about what Dave had to say. The blog, or my cocomment plugin, borked (yes, that's a technical term) and wouldn't let me leave the comment so I decided to share it here. You may like to read Dave's post before continuing with this.
Breathtaking post, or was it three? ;-)
I did the same exercise with my dept. We also had the same vastly differing results you did. At a provincial in-service about 9 or 10 years back I participated in the same exercise using real student generated work. Results varied from around 33% to 80%. This is one facet of Academe's Dirty Little Secret. Anyway, in my dept. we've been looking at how we assess all the content in all the courses we teach; one course at a time, one unit at a time. We're trying to develop a consistent approach to assessment at least within our building. We'll be "at it" for a while yet.
Fluent knowledge and recall of basic addition, subtraction, multiplication and division facts are essential for ANY student to experience success in math. I'm on the same page you are Dave.
A grade 9 student, who struggles (mightily) with her multiplication facts, and I were talking about this last week. As I was trying to help her I asked why she thinks I feel it so important for her to become fluent in her recall of the multiplication table:
"I know, I know, some day I might not have a calculator and I might need to multiply two numbers."
[Oy! Who tells kids this stuff? And do they really believe that? -- I mean the adults, not the kids. I know the kids don't believe that.]
"No. That's not why. You'll always be able to get a calculator if you need to multiply a bunch of numbers. That's not the reason. It's that you need to know the language of math so you can join the conversation."
"If your teacher is trying to teach you why multiplying pairs of negative numbers always have a positive result, or why, when we divide fractions, we 'multiply by the reciprocal' they're going to talk about stuff like 7x8 and assume you know it's 56 and go on to discuss some deeper ideas. If you're hung up on 7x8, need to pull out a calculator, you're going to miss the entire conversation. Your brain will be back 5 steps while everyone else is talking about this other stuff. By the time you figure out what's going on you won't know what's going on. You'll feel lost and confused and fall farther behind."
"Why do I need to know math anyway?"
"For the same reason you need to know how to read. Because it's a fundamental way that humans communicate with each other and understand the world around them. If you can read but you can't understand mathematics then there will be giant tracts of things happening in the world around you that you'll never understand."
[Whew! Went on a bit of a rant there. I'm going to get a cup of tea ... Cheers Dave!]