Today the 7th graders were studying integers. We are getting ready to add and subtract them. Yesterday we represented adding and simple subtracting on a number line. Today we went with a chip board, with black chips representing positive numbers, and red chips representing negative numbers.
I told them that we are going over multiple representations of adding and subtracting to get to the best understanding we could about the operations. I emphasized we are not going to depend on memorized rules, we must strive to understand adding and subtracting. This was of course met with some blank stares and confusion. During their previous year in 6th grade the teachers quickly skimmed the service of arithmetic with negative numbers. Of course they talked about rules like adding two negatives is always negative, multiplying two negatives is always positive and so forth.
Everything was going quite well with the chip board lesson for the first 30 minutes. Then with 14 minutes left I asked them to solve -4 - -3 and represent the solution on a chip board. This was the first negative subtracted by a negative problem of the day. The previous section and last section of the day handled this fine. They used the methods we had talked about and it resulted in a short discussion.
However in my middle section of 7th graders, most of the students had ended up with -7 as an answer. The seating arrangement of the class has 6 groups of 4-5 students in each group. Four of the groups had come to a -7 conclusion. One of the groups came to a -1 conclusion. The last group knew the answer was either -7 or -1 but they couldn't decide which way to go.
So we set up a little debate on the topic. A very loud and confident 7th grader proudly proclaimed a misstated rule from 6th grade and proclaimed -7 to be the answer. I held back any judgement and most of the students nodded in agreement. I then called on a student to present their argument for -1. They went to the smartboard and explained in a perfectly correct way that -4 - -3 was -1. No mention of rules they remembered they just used the chip board. When they finished 2 students loudly and confidently recited different rules incorrectly proclaiming -7 to be the answer.
At this point I reminded them how using rules can be confusing and we have to focus on what adding and subtracting means. This fell on completely deaf ears. At this point the 4 groups held fast to their wrong -7 answer, but there were now 2 groups solidly in the -1 camp. We were also down to about 4 minutes left of class.
Now I love student debate, I love discussions in math, I love to hear the thought processes of students, and I whole heartedly believe that mistakes push learning forward. However, I just couldn't bring myself to let the class leave thinking that -4 - -3 was -7. So I asked for anybody to argue for the -1 answer, because there were several students still shouting out arguments for -7. When nobody stepped up I felt I had to make an argument for -1.
So I asked them the questions that I thought would end the debate:
"If you have 4 red things, and you take away 3 red things, how many red things do you have left?"
They all correctly answered 1 red thing. The 2 groups beamed with delight that -1 was correct. Most of the other students looked a little confused, and 3 students starting misquoting rules to me about negative numbers to continue arguing for -7. At this point I told them that -1 was the answer. I showed them in a similar way to the first student using the chip board that -4 - -3 was -1. Some students still did not believe me.
At this point there is about 1 minute left of class and I am desperate to make sure students know that -4 - -3 is -1. So I made a last ditch effort and pulled up my smartboard calculator. I typed in the problem, asked them if it was entered correctly and hit enter. When -1 popped up as the answer, I actually think there were still students who thought -7 was the answer.
There is no great ending to this story yet. I have to tackle all the misconceptions tomorrow. I am pretty sure I completely messed up this lesson and students thinking about integers. I am not sure what I should have done differently after the debate started. I made sure in my next 7th grade section to take the subtraction slower and emphasize that subttracting is 'taking away.' Now I just have to figure out how to right the ship for a group of stubborn 7th graders.