“When you add a positive integer with a negative integer, how do you know if your answer is positive or negative?”

“Well if the negative number is bigger, then the answer is going to negative. If the positive number is bigger, then the answer is positive.”

“Aren’t all positive numbers bigger than negative numbers?”

“Well, yeah. But if you take the sign off the negative and it’s bigger than the positive, then the answer will be negative.”

“Why are you taking the sign off the negative number? What rule allows you to do that?”

“Uhh…”

“I know that I can give you 20 addition problems and you will probably get all 20 right, but I want you to explain to me why this works the way it does. Come talk to me when you think you have an answer.”

*10 minutes goes by*

“Alright, I think I’ve got it. If the negative number is farther down the number line than the positive number, then the answer is going to be negative.”

“Farther down the number line?”

“Yeah, it’s more negative than the positive number is positive.”

“How do you know that?”

“It’s farther from zero?”

“Oh, what do we call that when a number is farther from zero than another number?”

“Uhh…”

*5 minutes later*

“ABSOLUTE VALUE!. If the negative number has a greater absolute value, then the answer is negative. If the positive number has the greater absolute value, then the answer is positive.”

“That is correct young grasshopper. You have done well. You may now enter into the realm of proficiency.”

I have had this conversation about 10 times over the last few days. Our current system has students take a common formative assessment (CFA)which is very closely aligned to our state’s standards. It’s a multiple choice test that has questions that look an awful lot like the same questions they’ll be seeing in April when we take the CST test. Based on their score, they have a set of activities to do before they can re-assess. Re-assessment may look like the conversation above. I think I am really going to like this system because it allows for dialogue between teacher and student. I have the opportunity to ask them about the why and actually tie it to their grade. The benefit to this is that students have choice in how they demonstrate their proficiency the second time. The first time, it’s a multiple choice test. However, the second time may be written, oral or heck, they may even draw a picture. One of the best things about this is that the students are taking more ownership of their learning because they have to direct some of the activities. They actually have choice. And that’s empowering. They aren’t waiting for me to give them another hoop to jump through.

The parents are coming along slowly. Many of them didn’t understand how their student could score 100% on the CFA and yet the score in the grade book shows up as 80%. Last night was Back to School Night and I got the chance to explain that each standards’ assessment is two parts. The first part is multiple choice and the second part depends on the student. Once they realized that their child’s grade quits improving when they quit trying, I think they got it.

It’d be nice if we could focus less on the grade and more on learning, but…

…baby steps.

I prefer the zero pairs explanation, myself 🙂

I like your setup, though.

Yeah, I’ve tried the zero pairs, but haven’t been able to make it stick. I have found that the biggest part of the whole process is doing away with subtraction all together. We add the opposite up in here. Once I get them convinced of that, then they intuit the “bigger” number determines the sign. After that, it’s just vocabulary. These kids I’m talking to already know how to do it, I’m just trying to get them to put their verbal skills to the test.

I’d take this from a student:

If you spend within your means, you’ll still have money.

If you spend beyond your means, you’ll owe money.