This summer, I read the book, Building Powerful Numeracy for Middle and High School Students by Pamela Weber Harris. I actually read all three: the original, the worksheets/strings book, and the facilitator guide. I’m totally in love with the idea of them. I have been for a while, actually: ever since Kara Imm introduced me to them in the Development of Mathematics PD that I did with her two years ago now.
This fall, I’m at a new middle school (new to me), and I’m teaching a new grade (two in fact!). So I figured this was the perfect opportunity to introduce myself and my students to this new routine. I have fallen in love with the open number line, a tool I’d never heard of before Kara, but which features prominently in Harris’s book. I also have come to love the ratio table for multiplying and dividing, even more so than the array model, which I was already somewhat familiar with. To get acquainted with the new curriculum I’ll be teaching, I sat down and did the released problems from the NYS state exam for 7th grade (haven’t started planning 6th grade as much yet!), and I was absolutely floored with how many problems became easier using techniques I learned from the book. Especially the constant difference strategy for subtracting when used with mixed numbers and the idea of relating fractions to time.
But now comes the infinitely trickier part: figuring out when to use them in my own class and where to start. Via tweet, Pam recommended to me that we should use them 3-4 times per week, which means almost every day. I’m not sure what to expect from my new students in terms of proficiency with mental math, algorithms, and math in general. I know my new school screens students for admission using a math test, so typically, they’re at or above grade level, but that’s not a hard and fast rule. I don’t know anything about my feeder elementary schools, so I don’t know whether to expect the incoming 6th graders to be proficient with any of the models that will be used.
I do know (from talking with my new colleagues) that there’s a broad range of skills evident in the students and a broad range of affect towards math and self image (in terms of confidence as a math student). There are varying degrees of tolerance for anything perceived as “babyish” – and what’s considered babyish to some students might be fun or silly to another. So there’s definitely a delicate balance to be struck in terms of what strings to use, so the students don’t think I’m talking down to them or teaching beneath them (not my goal at all!).
For 7th grade, my first unit is about data and probability. So some of the skills students will need to do include taking the difference (range), summing a set of numbers & dividing (mean), and doing operations with fractions (probability includes adding and multiplying fractions). I’m wondering if we should start with a subtraction string that emphasizes the idea that subtraction can be the distance between two numbers (in addition to the take away model of subtraction). I’m seeing the connection there to the idea of the range. I’m also unsure of how to start here because the models and strings and strategies seem to all builds on each other: subtraction builds from addition, and so does multiplication. Division builds from multiplying and subtraction, and fractions, decimals and percent build on all four operations with whole numbers. So I’m not exactly sure where to launch with 7th grade.
In 6th grade, my first unit is integers, but we go in depth with them, covering 7th grade standards too. We do operations with them, as well as coordinate plane in 4 quadrants. I feel like to some degree, since we’re doing arithmetic already, the number strings should fit in more easily, but I have no idea what the proficiency of the incoming students is, so I have no idea what would feel too easy and what would be the right level of challenge. I don’t necessarily want to start with negatives (certainly not in the first set of numbers, as that should be accessible to all), but I think it’s the perfect place to draw comparisons between arithmetic with positive numbers and arithmetic with negatives. I also plan to write a whole other post on introducing them to negatives in general, because I’m not sure what the best method/model/context to use is. I’ve been doing a lot of reading about this topic already, and I can see that negative money alone is insufficient.
Anyway, I’m glad we don’t go back to school until after labor day. That gives me some time to reflect more on creating/choosing the number strings to use with my classes. I think it’s actually perhaps harder in some ways to envision where to start deploying this routine with curriculum that I’m unfamilar with. I guess I’ll ask you, the readers, to respond with any advice or thoughts, especially if you have familiarity with Harris’s work or have done number strings with your own classes.