I gave my three 7th grade honors classes a test last week on Patterns and Unit Rates. With some review problems – just order of operations and basic skills in fractions and decimals. I thought the review problems would help their grades. It turns out over 60% of the students made mistakes in their order of operations problems. Upon further investigation, I realized it was almost all due to errors with integers! Errors in adding or subtracting and multiplying. We didn’t have any division problems.
I started thinking about how we have been teaching integers in 6th and 7th grades – by teaching our students the rules. And making up fun “games” to help them remember the rules. Such as “when a bad thing happens to a bad guy it’s good!” I also thought of the 7th and 8th graders who had asked me after class for some help with integers – they knew that two negatives made a positive – so why were they getting – 17 – 5 wrong when they put 22? or 12? I remember doing this problem with one of the girls:
me: What’s your favorite store?
girl: Albertson’s (i thought she’d say Forever 21 or something like that)
me: Ok, say you bought something for $10 (I write -10 on the board)
me: Now you go to Baskin Robbins and buy ice cream for $6 (it’s by the Albertson’s)
Now the board shows: -10 – 6
me: How much have you spent?
girl: I know you change the minus to a plus because there are two negatives…
me: WAIT! You just SPENT $10 at Albertson’s and $6 at Baskin Robbins! How much have you SPENT?
girl: (looking at board) UMMM
me: Stop looking at the board and JUST THINK ABOUT IT. You’ve spent $10 and $6. How much have you spent?
We talked more about it, but I was remembering this conversation after I analyzed the tests. Something I didn’t say earlier is I’m the department chair now, so I have a little more clout and I think I can tell the other 7th grade and 6th grade teachers that we need to make some changes. I decided to talk about it at the next department meeting and say “We’ve got to take the time and let our students learn WHY integers work. Our students don’t know WHY anything works, we’ve been so busy trying to teach everything that we’ve developed great ways to help our students. We’ve given them the rules, shortcuts, ‘recipes’ to follow, and we’ve worked really hard doing it. However, what has happened is the students are memorizing how to do the problems, but they can’t tell us why these rules work. Now that we have the CCSS we can slow down a little, and really teach, then reteach if the students don’t understand. And we need to start now. With integers.”
Between the grading and the talking there was a lot of thinking on my part, and talking to a couple of other math teachers, and trying things out on my husband (who is such a good sport, and asks great questions). So by the time I talked to the teachers (about a week) I had put together an Integers Teaching Progression. I had started it with my classes and passed out copies to our teachers. I asked for them to please try it, and give me feedback on what went well and what should be improved, removed, extended, etc.. I don’t think it’s perfect, but it’s better than what we were doing! I’m including the dropbox link: https://www.dropbox.com/s/byz5jj5drrpzmof/Progression%20for%20teaching%20Integers.docx (Updated 10/14/13)
If you’d like to use it and give me comments on how to improve it that would be wonderful. I made worksheets to go with it because one of my teacher friends said she’d like them. So far they seem to be working ok, but it’s a lot of copying!
Ok, that’s it for my first blog post!