Differentiating in Math

A few years ago, when I started teaching fourth grade, our school's "policy" was that students who scored __% (it varied) on the beginning-of-year math assessments (which also varied), and were deemed "strong in math" by previous teachers, were moved up to the next grade level for math.  The third, fourth, and fifth grade teams had to align our schedules so that math happened at the same time in all classes, so that students could go to which ever math class they were assigned to without missing other subject-areas.  In a lot of ways, this worked for our school -- the kiddos who were moved to the grade-level above felt good about themselves, their families felt good about the "challenging" math instruction their kiddos would receive, and at least some teachers felt good about having all of the students in their math class working on the same grade-level standards. This method of "differentiation" was also pretty easy; we made the decision at the beginning of the year about who was in which math class, and it basically stayed the same all year.

However, it wasn't a perfect system. Other trends started to emerge. Within math classes, the range of abilities was still quite wide, but it was now students working well below their own grade-level mixed with younger students working one or two grade-levels above. The kiddos who were "left back" in their own grade-level math classes started to say that they were "bad at math."  The younger students who were pushed forward consistently out performed the on-grade-level students, often dominating class discussions, but also sticking to themselves during partner or group work.  Families of children who were not pushed forward but felt that they should have been complained that their kids were bored and/or not challenged, which sometimes came to a head when a student would be moved mid-year; although this also led to more questions about what qualified a student to move up and how and when.

Last year, when our school fully departmentalized fourth and fifth grade (and got a new principal), we decided to move away from this system.  We justified the shift for a variety of reasons, including that we had a new curriculum, which was more rigorous than our previous curriculum, and that there was new a district-wide focus on mathematical discourse, which we felt required students to engage with peers at a variety of levels.  We also felt that this change would help us address the culture around math that had developed in our school -- with kids thinking they were "bad" at math if they were working on grade-level.

To ease the shift, we also put in place a few opportunities for students to continue working in more homogeneous math groups within and outside of the math classroom. I held "lunch bunches" with the students who were essentially repeating fourth-grade math to set individual goals and reflect about their math growth. Another teacher pushed into my room two or three times per week to pull a small group of these "high kids" to work on challenge problems. Once a month we held "math seminars" where we split the kids across fourth and fifth grade to work on more critical-thinking problems at different levels.

I learned a lot from these experiences! It turned out (not actually surprisingly) that some students who were good at mathematical computation, and even interpreting basic word problems, really, really struggled with open-ended questions. Other students who had weaker computation skills, were very strong in reasoning. And many students struggled with communicating precisely about their mathematical thinking.  As a result, I introduced more problems that required reasoning, practiced math vocabulary to improve discussions, and looked for more ways to have students demonstrate their understanding about concepts.

Even as I'm typing this, it's hard for me to believe this was all just last year!  Thinking back about all of the questions and challenges we faced last year, I'm realizing how quickly some of the issues resolved themselves, while others we've still barely addressed.  I've come a long way in my own way of thinking about and planning for math instruction and differentiation. This year, I have a completely different group of kiddos, with different strengths and areas of need which has led me in different directions.  As I remember all the things I tried and learned last year, I'm realizing that some practices have already become second nature and many more are still very much a work in progress.

More on this soon 😉

Learning Long Division

Math was definitely not my favorite subject growing up... I had trouble remembering multiple steps to solve problems and never had a good idea if an answer was "reasonable." Nevertheless, one of my favorite memories of doing math in elementary school was creating and solving extended long division problems during indoor recess in 5th grade.  We would write up 25 or more random digits and then try to divide by 2 or 3 or 5, working our way across the entire chalkboard! It was so satisfying to simply "divide, multiply, subtract, drop down" and get a huge answer that I could feel confident was correct! I now realize that I had no understanding of why those steps worked or what that answer meant, but it felt "smart."

Now as a math teacher, I don't want my students to

blindly accept that a set of steps "just works" -- I want them to be able to explain how, why, and when any algorithm is useful and efficient. Last year our math curriculum intentionally avoided teaching long division in 4th grade to encourage students to use place value understanding and other strategies to divide.  I enjoyed teaching this way... I could explain why each method worked and observed students making logical connections between multiplication and division to solve complex problems.

This year, however, we are using Eureka math which does teach the long division algorithm (alongside other place value strategies) in fourth grade. As much as I loved using this method myself, I was anxious to make it meaningful for my kiddos. We began with mental math division (25 ÷ 5 or 18 ÷ 3), which required students to use multiplication facts to solve quickly.  I wrote the equations out using the long division symbol, but resisted reciting the "divide, multiply, subtract, drop down" mantra. Instead I asked the kiddos questions like "why do I write this here?" and "what should I do next?" and "what does this number represent?"

Then we added in using "place value disks" to represent the division (see above).  I demonstrated using the long division algorithm alongside the place value disks to help students see the connection. The kiddos totally got it!  My Teaching Fellow even commented how clearly she could recognize the steps of the algorithm within the place value model when it was taught this way!  

After two days of using the place value disks and word problems to provide context, it was time to go all in on using the long division algorithm!  Rather than keep everyone on the carpet for a traditional lesson, I allowed the kiddos to decide when they had mastered this new skill. When they felt confident with the algorithm (without drawing out the place value disks),  they could leave the carpet to start the Problem Set. Those who still felt stuck or unsure, stayed with me to keep practicing. I could practically see the gears turning in their heads and they pictured each step. When the lightbulb finally went off, they were so proud of themselves! 

Naturally, there were a few kiddos who were still struggling at the end of class. Rather than move on or wait another day to review, I offered to host a "Long Division Lunch Bunch." I ended up with 24 fourth graders eating lunch and doing long division in my room that day (I teach three classes so I offered the Lunch Bunch to all of them.) As the kiddos ate and worked together, I could see them gaining confidence!  By the end of lunch, most of them had the same satisfied, smart feeling I remembered from 5th grade. But this time, as they followed a set of steps to solve multi-digit long division problems, I knew they really understood what it means to divide one number into another and how those set of steps make their problem solving more efficient! 😄