In my most recent blog entry I described how even the very able students at Exeter, working in a problem-based environment, have trouble avoiding common misconceptions. I happened to see and cite an example concerning the distributive property.
I thought I would check out released test items. Here’s a 10th grade item from the PSSA in Pennsylvania:
Oops! Only 40% got it right. And there are many others like this. Here’s one from NAEP:
Now, I am sure there are readers who will sigh and say that these items are not sufficiently interesting/relevant/real-world, etc. and I will agree. But that’s not the point.
The point is that even after a year or two of algebra MOST students cannot use the distributive, (and often the associative, and commutative) properties properly. And that’s a problem with the INSTRUCTION, not the kids. Because the misconceptions are predictable; because it takes a lot of iterations to overcome what is counter-intuitive about much of higher-level math, you have to keep probing for this understanding – as the Exeter – and the Harvard Physics – example so clearly showed us. But because conventional textbook coverage is so fractured, unfocused, superficial, and unprioritized, there is no guarantee that most students will come out knowing the essential concepts of algebra.
Don’t you math teachers get that there is a problem? The ‘yield’ from your ‘coverage’ is terrible. So, clearly, ‘coverage’ is not the key to optimal performance on tests. Some day we’ll know why so many math (and history and science…) teachers think coverage is optimal preparation for tests.
PS: The NAEP Question Database is here.