Here’s a conversation between collaborators and friends Kara Imm (Co-Director, Math in the City) and Geoffrey Enriquez (Math Teacher, Vanguard High School), who both work and live in New York, NY.
Geoffrey: Hey, Kara.
Kara: So, you and I have been talking about issues related to numeracy for high school kids for awhile now. Recently, in your capacity as a Master Teacher at Math for America, you created (with our friend Marcelle Good) a PLT — professional learning team — for teachers, focused on the development of number strings.
So, tell the people, what’s a PLT exactly? And why only number strings?
Geoffrey: Developing numeracy with students is often a challenge for high school teachers — rigid pacing calendars, testing, and inconsistent exposure to numeracy routines used by K-8 teachers. A PLT focused on numeracy building was the best way to address this gap in pedagogical knowledge. Our number strings PLT was a working group of teachers from a broad range of grade levels (K-12) and schools who met monthly to develop and implement number strings in their classrooms.
Kara: As you were thinking about the development of teachers in this community, what did you hope they would appreciate about number strings?
Geoffrey: Our intention was that participants would walk away with at least one number string that they developed — linked directly to what students were learning or doing in their classrooms. We hoped that participants would be able to learn the characteristics of a number string and use them daily either as short routines (middle school/high school) or lengthy class discussions (elementary/middle school).
Kara: So, as a teacher of teachers, how did you structure the PLT so that people could become local experts in strings? What kinds of activities were really helpful for the group?
Geoffrey: It was a working group where participants would be able to improve in their practice in implementing number strings. During each session, we asked 3-4 volunteers to bring in artifacts from their classrooms to share and receive feedback. We adapted a Looking At Student Work protocol to create a safe space for educators to share, wonder, and discuss their work — in hopes that they received ample feedback from other teachers.
We then asked participants what topics (fractions, negative integers, multiplication, proportions, etc.) they were interested in working on so that they would be able to work in pairs or groups on developing a number string. We devoted some time to practice leading our strings with a partner. We were pleasantly surprised that everyone who volunteered to share work (different teachers every time) brought in work and were willing to share what was happening in their classrooms.
Kara: As you know well, a real challenge with high school kids is that this is not the first, or second, or even third time they have “seen” this mathematics. So we want to be really careful not to treat them as if they were 4th graders, constructing this knowledge for the first time.
How did you respond to this issue — Did you draw upon strings for younger students, or did you need to design entirely new strings for your students?
Geoffrey: Yeah, it was a challenge for us — we had to modify and design new strings for students based on the grade level or performance of each of our different classrooms. Teachers who worked in elementary and middle schools found a lot of common ground. High school teachers had to do a bit more thinking; it was difficult to think of contexts for situations involving exponents or fractional powers, for example.
Kara: When high school teachers talk about their kids “not having” any number sense, I often re-frame it as needing more time, and new (and different) ways, to construct the same knowledge that other kids have. Thankfully, we don’t all learn in the same way, or at the same pace — but all kids are entitled a set of experiences that will allow them to construct knowledge deeply. The challenge for high school teachers is to acknowledge that prior attempts at teaching have happened, but that they weren’t enough or they weren’t the right set of experiences for that student.
You’ve been teaching in New York for 11 years now, and you’ve witnessed the kinds of numeracy and algebraic needs of high school kids. Have you found this to be true? What role have number strings played in addressing this dilemma?
Geoffrey: A few years ago I was talking to two of my 12th grade students during lunch about a numeracy workshop I attended the night before. Chekeshia, who enjoyed math class, shared how she often adds in her head by rounding/using 10s (something like 42 + 39 becomes 42 + 40 – 1) while the other student, who struggled in class, shared how she visualizes the “stacking” algorithm.
This got me thinking that students who “hate math” probably had bad experiences where numeracy became replaced by procedures and algorithms. Students are often “addicted” to algorithms like stacking, cross-multiplying or memorizing steps to solve an equation in place of sense making. Implementing number strings in ~15 min daily routines have helped to use students’ mathematical intuition and infuse them into making sense of problems and concepts.
Kara: Oh, we love cultivating mathematical intuition. That’s really lovely.
Let’s talk about those folks who are not yet convinced that these routines are worth their time. The critique from high school teachers is that number strings are so wonderful for younger kids — when they are first making sense of these ideas — but there is just too much content to teach at the high school level to devote time to these routines.
My question — What would you say to teachers who hold this belief?
Geoffrey: I find that my high school often has high performing graduates who end up taking remedial math courses in college because of low scores on placement exams. In most cases, students tell us that fractions, decimals and radicals show up so much on placement exams and they wished that they practiced these more in high school.
From my own experience, I’ve noticed that middle school and high school teachers often avoid tough numeracy (fractions, decimals, negative integers) — changing numbers in problems so that “nice” positive integers are used in the process of solving a problem. It seems like a good practice — if your goal is to teach students how to solve an equation, why have them get stuck when they see fractions or decimals that might take an extra 10 minutes of class time? But we don’t live in a world of positive friendly integers. Avoiding “tough numeracy” prevents students from developing their number sense. I would ask teachers to think about these questions before they worry about what is on the state exam.
Additionally, I would ask teachers how they build in sense making and equivalence into their classrooms. Number sense can help to address common misconceptions including 2x + 2x = x2 or (x2 + 2x + 5) / x = (x + 2x + 5). Using number strings and sense making can help to address these issues. Is 2(3)+ 2(3) = 32? Or is (22 + 2*2 + 5) / 2 = (2 + 2*2 + 5)?
High school teachers often teach algebraic concepts but can use help in developing ways to have students prove or disprove if expressions are equivalent. I would ask high school teachers why our established “rules” (e.g., subtracting/adding the same number from both sides of an equation, multiplying a negative by a positive is a negative) are true and how they can use this practice of understanding number structures to deepen student knowledge.
Kara: That’s really interesting. I think taking the “long view” of learning — as you know we often do at Math in the City and in this blog community — allows us to see how the choices we make as teachers, at any grade level, might impact the trajectory of kids’ learning. Ideally, ideas are always in development, connected to something that came before and preparing us to make sense of what is yet ahead.
Geoffrey, this was fun. I hope you will share with us some strings that you and your MfA colleagues wrote and we can keep talking.
Geoffrey: Of course!