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Instructional learning coach Julie Everett analyzed math assessment data over several years in the five elementary schools where she works in Kearney, Nebraska, and kept noticing a persistent problem: number sense was lacking. Many students did not have basic understanding of the relationships among numbers. They did not know how to think or talk about numbers or use number sense reasoning strategies to solve problems. Without those foundational skills, Everett believed students would likely struggle in higher-level math classes.
She discussed her concerns with colleagues, and then in Spring 2013, Everett discovered Jessica Shumway, author of Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 (Stenhouse, 2011). Everett heard Shumway present at the National Council of Teachers of Mathematics annual conference and knew she had found a valuable colleague and resource.
"I was highly impressed with her background knowledge and the research base she had done," Everett said. "I had several conversations with her, and discussed how we might involve her in consulting with the district. Our curriculum and instruction team believed that we needed to be doing something more systemic and systematic with math and literacy and improving instruction with our teachers.'"
Over the next few months, they developed a three-year plan to help all elementary teachers in the school district learn the techniques that Shumway shares in her book and in her new DVD, Go Figure! Number Sense Routines that Build Mathematical Understanding (Stenhouse, 2014). Their plan started with a book study involving the kindergarten and first grade teachers, expanded to include Skype sessions with Shumway, and finally led to on-site visits where Shumway modeled routines and cotaught small- and whole-group lessons with the faculty.
In addition to reading the book, the Kearney teachers also had to write a personal reflection every month, sharing what they had learned from Shumway's book and what they were doing differently in their classrooms as a result. Everett believes the requirement made teachers more accountable for synthesizing information and focusing on results.
"It's just been a really cool experience," Everett said. "At first, I have to say, our K-1 teachers were overwhelmed by the work that was expected: 'We have to read every month? What is this all about? We don't have time for that.' There was grumbling at first." But after Shumway showed the strategies in application and helped teachers take risks and raise their expectations, Everett said, "I would have to say that 80-90 percent of our K-1 teachers have now said, 'Wow, this has totally transformed my thinking about math. I had no idea number sense was so critical.'"
The Importance of Number Sense
As Shumway relates in her book, teaching number sense is not only critical, it's also complex. "There are many layers to it, and it is rooted within all strands of mathematics," she writes. "Number sense facilitates problem solving, reasoning, and discussing mathematical ideas." Students with strong number sense can visualize quantities and perform mental math, understand the relative magnitude of amounts, make comparisons among quantities, and determine the reasonableness of an answer, among other skills. "Embedded in these characteristics of number sense are big mathematical ideas; strategies that utilize number sense; skills, models, and tools for using number sense; and language for explaining number sense ideas and strategies."
Just as athletes stretch their muscles before every game and musicians play scales to keep their technique in tune, mathematical thinkers and problem solvers can benefit from daily warm-up exercises. Shumway has developed a series of routines designed to help young students internalize and deepen their facility with numbers.
Shumway also shows teachers how to move students through what she calls the Early Number Sense Learning Trajectory, starting with subitizing, understanding magnitude, and counting and progressing to hierarchical inclusion, part/whole relationships, compensation, and unitizing. The goal is to develop children's flexibility and fluency with math. Shumway says these methods involve teaching the meaning of numbers, rather than procedures and memorization, so that students are able to decompose numbers, visualize them and apply them in the future.
"Think about it in terms of reading," she writes. "It is cumbersome and inefficient to sound out every letter in a word. When children begin to recognize and use chunks of letters within a word or read sight words, they become more fluent readers. This frees up their cognitive energy for more challenging words. It is the same in mathematics. Seeing groups and thinking about amounts in terms of groups leads students to become more fluent and numerically literate. Their cognitive energy can then be spent on more challenging problem solving."
The Urgency of Understanding Math
For many elementary teachers, Shumway has instant credibility. In addition to having worked as an elementary teacher and math coach, she acknowledges having had weak preparation and understanding of how to teach math. A history major in college and a "social studies guru" when she began teaching, Shumway had to deepen her own knowledge of mathematical thinking along with her students.
"She is clear to say, "I am not a mathematician and I always felt that I was poor in math and that it was because of a lack of number sense. And what's why I have an urgency to make sure that teachers understand the very important piece of number sense and why that leads to success for kids,'" Everett says. "Teachers could relate: 'Oh, this is me,' or 'that's how I feel.'"
As part of their book study of Number Sense Routines, Kearney's kindergarten and first-grade teachers had to choose three students to follow in a case study through the school year. They set goals for all the students, tracked their progress in math, and shared the information and consulted with their colleagues each month. If a student achieved the goals set for him or her before the end of the year, the teachers selected other students to follow.
"It was fun to hear teachers talking about those kids," Everett says. "They would ask, 'How is Karl doing?' It became very personal. We had never had those conversations before. The sharing piece is just so enlightening and refreshing. It becomes a problem-solving event, as well as a celebration of moving kids along their learning continuum."
Teachers also taught demonstration lessons, with Shumway observing and deconstructing the vocabulary they were using with students and the questions they were asking. They observed and taught with teachers in other schools.
"I feel it's important to talk to other teachers outside your building. It gives you more perspective," says Marissa Schleiger-Kruse, who just finished her first year teaching first grade at Buffalo Hills Elementary in Kearney. "I feel that everyone, new or old, has benefited so much from Jessica Shumway and her Number Sense studies."
Schleiger-Kruse says she incorporated many of the practices in Number Sense Routines, including one called Count Around the Circle, which helps students understand the pattern the teacher is describing, such as counting by twos or fives or counting backwards.
"I've done those every day, and it helps students learn their counting routines," she says. "Eventually some of my higher learners, I know that it will help them with multiplication because it's really skip counting. If we practice that daily, they get that."
The consistent practice benefited every student, Schleiger says. By the end of first grade, her school district expects students to be able to count by twos to fifty and count by fives and tens to one-hundred.
"All of my students have mastered that, and all it takes is five or ten minutes a day," she says. "We may start at 200 and count backward by fives or tens. They love it; it's never boring to them. They are always trying to figure out what I'm going to start with."
Central Elementary School first-grade teacher Tara Abdallah says she and her colleagues appreciated the practical strategies and tools Shumway shared that they could immediately use in their classrooms. One of her favorites is Dot Cards, which resemble domino tiles or dice and help students practice skip counting and recognize groupings and multiples that they can later form into equations. The visual aids help students learn to subitize, but they also let teachers continually assess how students are thinking about amounts.
Kearney teachers adapted some of the strategies in Number Sense Routines for other purposes and for subjects other than math. For example, Abdallah tweaked Shumway's Count Around the Circle strategy to help students learn to count money, and her coteacher adapted it for guided reading. Instead of using numbers, she substituted the alphabet and phonetic sounds so students could become more fluent when reading.
"There's some amazing stuff in this book that's so hands-on and freeing," Abdallah says. "I cannot wait until next school year. I'm going to implement so much from the book!"
During the 2014-15 school year, Kearney's second- and third-grade teams will begin the training cycle with Number Sense Routines, and the following year teachers in fourth and fifth grades will get involved. Everett says she hopes that teachers will keep the momentum going in future years, coaching their colleagues and planning collaboratively so that they can eliminate instructional gaps from one grade level to the next.
"Teachers are sharing way more than they ever have in our staff development," she says. "This is powerful. I am super proud of the work our teachers are doing."