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:
Results? Grim:
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.
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2 Responses
Your point is well taken, and it’s certainly a national discussion right now. Here is a great video of David Coleman reviewing the “shifts” within the Common Core Standards: http://engageny.org/resource/common-core-in-mathematics-overview/ My favorite “shift” is coherence. We need to spiral the same concepts throughout kids’ experiences in the context of meaningful problems. Thanks for sharing this.
What I see as the problem is that the students never learned how to solve or never had enough practice solving how to add and subtract fractions with different denominators. You do not have to remember the algebraic method to get the right answer to either question. You can just choose a number for X and then look at the answers and see which one fits the answer you’ve already calculated. This will reinforce to the child that the equation makes sense and is not just something that they have to remember.
I have been out of school for many years. I am not a mathematician, but I love math. I was taught the old fashioned way; what many would describe as drill and kill. I did not own a calculator until I got to college. I did not always understand the “theory” behind why I was doing this or that in math when I was in elementary school. I didn’t need to. What I was prepared to do is focus on Algebra when I got there; not try and learn the basics at that time.
The equations only make sense because you can do the basic math. I say show kids both ways, but make sure they can DO subtraction and addition and multiplication and division of fractions and then they will get ALL of those math problems right even if they forget how to reduce the equation algebraically.