For those of you involved with my twitter conversations, you know my students were really struggling with subtracting polynomials. Out of 36 students I had to pull out about 10 of them to work with individually.
It took me a while to figure out their misconception.
Say the problem looks like this (3x - 8 + 2x) - (4x - 3 + 6x).
I taught them to change the sign in the middle and flip every sign behind it.
Here's what they would do: (3x - 8 + 2x) + (-4x + -3 + 6x).
In middle school they were taught to change every minus sign to plus a negative. When I told them to flip the sign, they would, but they added in the negative which doesn't flip the operation. And even though I stressed every sign about a million times, they would always politely skip over the plus signs. There were other errors too but this was the most common misunderstanding.
With those 10 kids, I pulled them out individually and tried to explain and re-explain what to do and why (I just now realized I should have said to flip the operation instead of flip the sign). Later I realized it was even more confusing because it looks like they are changing the minus sign in the middle to plus a negative. Egad!
Too late I asked the other math teacher what I should have done and he suggested just practicing lots of problems like -(7x - 2 + 4x) so students can really see over and over how the negative sign is distributed.
Anyway, to get to the point of this post...I pulled students out and we went over their mistakes on the quiz. Then I gave them new problems to practice. Then I give them a new quiz. Quite a few students got a 100 on their retest. Some needed to do it three times instead of two but still got the 100. Now I feel guilty. I feel like it's cheating. I feel like I should have pulled out every single students who didn't get a 100 instead of just the ones who did terrible. Shouldn't every student have the opportunity to make a 100? I don't know. It's kind of like forced sbg. I took away their initiative to improve and forced them to 'want' to do better. I practiced with them and we conversed and we identified misconceptions. All the perks of sbg, right?
So why do I feel so guilty?
On my assessments for each method of systems of equations my classes did really well. But maybe we had just practiced enough times that they had the process memorized. How do I know that they truly learned? I guess that just means I need better assessments. How do you assess understanding other than having them show by doing or show by explanation?
I am afraid now that my administration might look at the grades and assume that I am not doing my job or not doing anything for my students to all have high grades. Does that mean I need to make my assessments harder so that more students will fail? Will I feel less guilty if my grade distribution looks more like the 'normal' bell curve?