By choice, I am designing and teaching a class for 8th graders who scored below 50% on their 7th grade Pre-Algebra final exam. It’s tracking. Sort of. I met individually last year with every student who was recommended for this class. Each student currently in the class chose to be there. They and their families signed a paper requesting this class (the alternative was a double period of algebra).
And I could talk your ear off about why I believe that creating and teaching this class was an equity decision. And some of you would talk my ear off about why tracking is bad, period. And I would disagree. We’d then have a really heated, informative debate where we’d both push our thinking deeper and become better educators because of it. Sometime soon, over a bottle of wine, the math department at my school will come to my home and we’ll have this very debate. It needs to happen. And as I said, it will make each of us think more reflectively about what we do.
But for now, here’s what I’m excited about: teaching integer addition. Mastering it is critical for success in Algebra 1. All of my Algebra Scholars took Pre-Algebra last year and had extensive experience with integer operations. Nevertheless, on an integer pre-assessment given shortly before I began my unit, only three students could successfully add one digit integers, and only 1 student could successfully add integers with larger addends.
I believe that they have been introduced to far too many models for integer addition, and never been given the time to learn any one model in depth. For this reason, the ONLY model we use in class is a number line, but we have taken a lot of time to have rich activities and conversations about how to use it to model problems involving integers.
I do wonder, though, if using multiple models is useful. I’m curious what others think.
Last year, I realized that when I do math, I love to use colors. It helps to organize my thinking and highlight important discoveries I’ve made while in the middle of a problem. So as of September, pencils are no longer required (though I make them readily available to students and allow them to borrow my supply at any time). Upon entering class, students take these which is a set to share with their neighbor. They are required to use colors when we take notes. They are not allowed to use them for doing their regular work, until I tell them that they should color code. I am trying to preserve the ink for as long as possible. They know that if I find a single dot of this extra-fine German ink on their skin, I will cut off that joint. Ok, not really, but I have been very clear about the purpose of these uber-expensive markers. Having these markers makes students (and teachers) not in my classes quite jealous. And since I teach the lowest ‘track’ of students, I like it that way.
Phase 1: Comparing Integers. Using Wu’s definition that a larger number is always to the right of a smaller number on the number line, we placed integers on the number line and compared. To escape bordom, each student had a plastic popcorn box filled with integer tiles from -10 to +10. They chose 3 numbers from their box and compared them on the number line. They then wrote 3 inequalities comparing their numbers. This pushed them to consider 4>-2 as well as -2<4.
Each student made a key at the top of their paper, color coded, using three colors: 1 for positive values, one for negative, and one for zero.
Phase 2: Adding Integers. Sticking with the popcorn boxes to generate random pairs, students then chose 2 numbers and added them. Everything was color coded and they asked themselves over and over again, “Will I make it to zero?”
Phase 3: Discussing discoveries. Students then took all their addition problems and sorted them onto a mat. I had no intention of telling them rules, but instead wanted them to really be able to ‘see’ in their heads what happens when you add integers. The colors helped to imprint the math in their heads. We had a rich conversation about what the colors showed us on our mats. After analyzing why positive +negative sometimes is positive, sometimes is negative, and sometimes is zero, I felt that well over half of my students had solidified their understanding of when you combine the absolute values (adding) and when you find the difference.
(Apologies for the head tweeking here):
Phase 4: Tweaking the model, but only slightly. I realized that asking oneself what the sign of the answer would be was a critical juncture in what should be happening in their heads. Most were really good about asking themselves, “Will I make it to zero?” But I needed a hook, something silly, which tapped into their teen-ager-ish-ness. This whole unit could be summarized by a single question, spoken in my sexyiest possible voice, “Hey Baby…What’s your Sign?” Our final push for mastery of large numbers involved asking each other this question, and stiffling the giggles that ensued. I took away the bulk of the scaffolding (most students had tired of 2 colors on the number line and ironically were begging to simply do their work in pencil). But added one last element for those who needed it: colored dot stickers. A few students still needed a visual reminder to cue them on when to combine the absolute values (add) and when to take the difference. They all had green for positive values and red for negative. 100% of them could now quickly and efficiently respond to the question of their peers, “Hey Baby…What’s your Sign?” And before doing any calculations, they stuck a flourescent orange sticker down with the sign of their answer. It took three weeks to get here, but we made it and I never gave any rules for them to memorize. Success. The real goal, of course, is that they conceptually understand this well enough to use it throughout the year and in the years to come. Stay tuned…Next week’s project is for them to design a series of lessons to teach to the students in the algebra support classes who are still struggling with integers. They can’t wait!