Math = Love: Remediating Integer Operations

Sunday, September 20, 2015

Remediating Integer Operations

First, a letter.  Then, actual classroom stuff.

Dear MTBoS,

Thank you!  Thank you for inspiring me to become a math teacher.  I always wanted to be a teacher, but you showed me just how fun teaching math could be.  Thank you for helping me survive my student teaching experience.  Thanks for encouraging me through year one.  And year two.  And year three.  And, now, year four.  You've answered my questions.  You've helped me keep my sanity.  I've stolen more ideas from you than I can count.  If it's happened in my classroom, there's a 99.5% chance the idea came from one of you.  Thank you for allowing me to meet my amazing fiance.  You guys aren't my "internet friends" anymore.  You're friends.  Plain and simple.  And, thanks for helping me make it through grad school.  For my most recent class, I had to plan a hands-on unit that was composed of multiple, linked learning cycles.  This sounded daunting until I started stringing together stacking cups, barbie bungee, and tying knots!  You've taught me so much more than any master's program ever could.  MTBoS, you've truly changed my life!


Now, let's talk some more grad school.  I just finished implementing my action research project in my classroom.  The topic of my project is remediating integer operations.  We all know how important it is for students to be able to work with positive and negative numbers in algebra and beyond.  When students come in our classrooms lacking these skills, it impacts their performance on a daily basis.

Disclaimer: I don't wholly embraced timed tests.  But, when you're trying to measure accuracy and fluency with integer operations from a large number of students on a regular basis, you do what you have to do.

Here was my plan:

Have parents sign a permission slip.  I did this project with all of my Algebra 1 and Algebra 2 students.  

Give students a pre-test.  50 integer operations questions.  5 minutes.

Grade pre-tests.  Input scores in excel.  Sort scores from highest to lowest.  Take the top two students.  Randomly assign the first student to one of two treatment groups.  Place the second student in the other group.  Take the next two students.  Randomly assign the first to one of the groups.  Place the second student in the other group.  Repeat until all students are assigned to a group.

I had my student aide help me grade a lot of these integer tests.  Her mind was blown when I folded over the answer key to make grading easier!

Give students a survey to help measure their attitude toward math.  Hope this comes in handy when you write up your findings.

Prepare 2 different sets of notes.  One set for each group.

Group 1: Just the Rules

Group 2: Number Lines - These were adapted from a resource I found online and have subsequently lost.  The notes featured printable pi shaped playing pieces to practice adding/subtracting on the number line.  I wrote my own notes for multiplication/division because I couldn't find any that focused on number lines.

Students in the second group also got a number line and a bingo chip with an arrow drawn on it.

Bingo Chip in Action:

My project requirements meant that students in each class period needed to be in each treatment group.  Therefore, whole-class remediation was out.  The remediation process needed to be self-guided.

Students also added a horizontal number line to their interactive notebooks.  I haven't actually glued mine in yet, so I can't take a picture of it.  But, students were instructed to cut out the rectangles including x's.  The remaining tab glues in the back cover of the composition notebook.  The number line then folds down and in to be hidden when not in use.  If you're confused, here's a post about the vertical number lines my students glued in last year to help them with slop.  

Are these notes perfect?  Absolutely not.  Having had students use them, I see so many ways to possibly make them better.  Use your judgment.  You know best what your students need.

On Tuesdays, Wednesdays, and Thursdays, I set the SMART Board timer for 5 minutes and passed around the flash card bucket.

Each bag has 48 flash cards.  These 48 problems were taken from the 50 question pre-test.  Students work through the flash cards while referencing their notes sheet.  Students with number lines should use the number line and chip to act out each question.  As students gained confidence, they only used the manipulatives on questions they struggled with.

I chose to use flashcards instead of a worksheet or other activity because of the fact that they allowed students to check their answers themselves and ask for help when needed.  When I went around to answer student questions, I always referred them back to their note set as I explained.  I told them that they should be asking fo

I blogged more about these flash cards and how I made them here.

On Fridays, students took the pre-test again as a check-up.  Tests were graded.  Scores were recorded.  On Mondays, students got their tests back to correct.  Students used their notes to rework any questions they got wrong, and they used the rest of their daily 5 minutes to complete any problems they didn't get to.

Monday - Redo missed problems on test
Tuesday - Flash cards
Wednesday - Flash cards
Thursday - Flash cards
Friday - Test

After doing this for three weeks, I wrote a post-test for students to take.  Students took the regular test on a Friday.  On Monday, I gave them a test with new questions.  The purpose of this was to ensure that students had not somehow memorized the 50 answers to the pre-test/weekly check-up test/flash cards.  I told them if they had memorized the answers to all 50 questions in order, I would be very impressed!

I graded Friday's test and Monday's test and put the grades in my excel file.

I typed up a short reflection sheet and used mail merge to import each student's scores for each test.  I typed up this form rather hastily, and it shows.  Oh well...

I was a bit disappointed in how little detail students used in answering the last two questions.  :(  I was hoping I'd be able to pull some good quotes from there to use for my research project!

So, I've got all my data compiled now to determine if the use of manipulatives increases the effectiveness of remediation of integer operations as compared to rotely memorizing rules.  One of my classes focuses on how to analyze this data.  I could try to start it on my own now, but I've got enough other stuff on my plate.

I found it interesting that none of my students seemed happy about what group they got placed in.  The students with number lines thought they were dumb and annoying.  The students without number lines felt cheated.  It absolutely broke my heart when I heard a student claim that I had only given him a number line because he was a special education student.  I want my students to see a number line as a tool that they should all be using.

Many of my students who weren't in the group that received the number line instructions still made use of the number line on the wall on occasion.  They asked if this was okay.  Absolutely!  My project just dictates what instructions I give you.  They were welcome to solve the problems in whatever way they chose.

Oh, and I guess I should specify this: scores went up!  In both groups.  I'll update you after performing further analysis.  In other news, only 4 more classes left of my master's program!

Download files here!


  1. A quick thought about the responses you got on those surveys - in my undergrad assessment class we talked about being specific about the length of answers. So "In 2-3 complete sentences, exaplain how...". It doesn't work with EVERY student, because some don't read directions, just don't want to, etc, but I have found that I am much more likely to get more in depth responses if I'm specific on length and complete sentences.

    Also, I wouldn't sweat the kids feelings on it too much. Research projects in real life often weird b/c the kids know it's not 100% authentic. My master's project, while great on paper, totally bombed in an actual classroom. It was so different from the norm that they never really bought into it. The important thing is that you learn something from it (and goodness knows I did!) Good luck with everything!

    1. Thanks for the feedback! I definitely should have been more specific about the type of answer I was looking for. I have learned so much through this project!

  2. Here's a number line board game that teaches kids how positive & negative numbers work:

  3. I am using this to help with remediation of my new students who did nothing beyond calculator use last year. I am not doing it as split classes but rather showing both the rules method and number line method. I just cannot wrap my head around dividing when the first number is negative. I have followed your steps dozens of times. Obviously, I am missing something? Everything else works great and is easy to understand. On - divided by + I am getting +. On - divided by negative, I am getting negative. Help!

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  5. In my humble opinion, there is a mistake in Sarah´s notes. Im trying to fix it. Anyway, Thanks for your amazing blog. Greetings from Spain.