We continue our Profiles in Effective PD series with a visit to Kearney, Nebraska, where teachers are in the middle of a three-year plan to implement the techniques discussed in Jessica Shumway’s recent book, Number Sense Routines. Stenhouse editor Holly Holland recently talked to instructional learning coach Julie Everett and shares how teachers in kindergarten and first grade are helping their students improve their number sense.
Teachers Get Fit with Number Sense Routines
By Holly Holland
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 SenseRoutines 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.”
Students who are lacking number sense tend to struggle in mathematics and the gap in their understanding can widen over time. In her book, Number Sense Routines, Jessica Shumway introduced a series of short warm-ups for the beginning of math class to help students internalize and deepen their facility with numbers.
Now in her new video, Go Figure!, Jessica invites you into three elementary classrooms for a live look at how these routines work. Viewers will see Jessica and three classroom teachers teach six routines:
• Count Around the Circle (2nd grade)
• Quick Images in a Guided Math Group (1st grade)
• Counting on the Number Line (4th grade)
• Ways to Make a Number (2nd grade)
• Visualizing Quantities (1st grade)
• Count Around the Room (4th grade)
Each lesson includes reflections and debriefs that unpack the teaching moves demonstrated in the classroom. In a concluding segment, Jessica answers nine frequently asked questions about number sense routines such as “Where do you get your ideas for routines?” and “What if students have difficulty explaining their mathematical thinking?”
Go Figure will be released later this month, and you can preview three segments (including a full routine lesson) as well as download the Viewing Guide now.
We wrap up our four weeks of math quick tips with Jessica Shumway today with tips and ideas for data collection routines in the classroom. You can still preview Jessica’s book Number Sense Routines on the Stenhouse website. Read Chapter 6 for more ideas on collecting data over a long period of time with your students.
Data Routine Tips, Ideas, and Questions Assigning Data Collection Jobs
It is important to allow students some element of choice for their data collection jobs. However, for management purposes, in the beginning of the year I choose their jobs for them based on skill level and needs (they indicate first, second, and third choices, and I take their requests into account). This allows for differentiated instruction through pairings. Students who are comfortable reading the thermometer, sunrise/sunset data, and moon phases data and recording that information hold these jobs in the beginning of the year in order to get the routines going. During this time, the student I assign to be Data Assistant is often not as comfortable with these skills, but the job of Data Assistant gives him or her time to observe and learn how to do the other jobs and pushes his or her learning to an independent zone.
During the second quarter of the school year, I often flip-flop the roles. I often assign the role of Data Assistant to someone who is strong in collecting and recording the data, and that person can assist the others in learning their jobs. This pushes the thinking of all the students involved. This pairing challenges those who are not yet proficient with collecting and recording data and it challenges the Data Assistant to explain his or her thinking; the Data Assistant is not allowed to do the other jobs for his or her classmates; he or she must use words to describe what to do.
Questions for Differentiation
• What do you notice about the data? (Use an open-ended question like this to spark discussion and give you a sense of where students are in their thinking.)
• Why do you think that? (Use a question like this in response to statements such as “It’s getting colder”; you are asking
students to talk about the data and cite evidence; you are asking them to “prove it.”)
• What was the temperature on October 10th?
• What days during this month were the warmest? The coldest? The most mild?
• What was the lowest temperature this month? The highest temperature?
• What has been the range of temperatures this month? (Emphasize the strategies for figuring it out by asking How do you know?)
• What days so far this year have been warmer than today?
• How many days so far this year have been warmer than today? (Again, emphasize the strategies for figuring this out by asking, How do you know? Some students might count one by one, some might group and skip-count, some might use the number of days in school and subtract, and so on.)
• What do you think the graph will look like next month?
Summarizing the Data Each Month with Mode, Average, and Range
• What is the most common temperature this month? (Mode)
• What is the mean temperature for March? (This arithmetic average/mean question would be appropriate for some third-grade students and many fourth-grade students.)
• What was the range of temperatures this month? (This question asks about the difference between the highest and lowest temperatures.)
Our math quick tips continue on this sunny Tuesday with another number sense routine by Jessica Shumway, exclusive to the Stenhouse blog. Her new book, Number Sense Routines, is still available for full preview on the Stenhouse website!
Using Algebra and Arithmetic Routines to Improve Number Sense
What goes in the blank?
Pose this problem to your students. Many of them will write 14 in the blank. Some will add 7+7+8 and put 22 in the blank. Others say that the equation is impossible. Some might answer 6.
During a Cognitively Guided Instruction training, my math coach Debbie Gates challenged me to present this problem to some fifth grade students. I was surprised that many of them wrote 14 or 22 in the blank. Some of them wrote 7+7= 14+8=22. Through their elementary school years many of these fifth grade students had developed misconceptions about the equal sign and what equality means.
A couple of years later I was part of a study group that read Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Carpenter, Franke, and Levi. It really got me thinking deeply about students’ misconceptions about the equal sign as well as the critical importance of encouraging students to think relationally. I saw these factors as critical to my students’ number sense development, so of course, we made it a Number Sense Routine in our classroom.
At the beginning of the school year, I simply started with a series of True/False Statements that Carpenter el al. suggested in their book (see page 16 of Thinking Mathematically):
I wrote these equations on the board (one at a time), and the students discussed whether each statement is true or false and, of course, explained how they know. This was the beginning of our conversation about equality and what the equal sign means. Many students believed that the second and third equations were false! I recorded students’ ideas about equality during the course of our discussion:
We did similar series of equations like these over the next two weeks. Other examples include:
True or False?
True or False?
True or False?
As we worked through these daily routines, we continued our discussions about equality and the meaning of the equal sign, but students also began to dive into important ideas like the commutative property of addition and multiplication, how to compose and decompose amounts, and relationships among each side of the equal sign (relational thinking). This was exciting!
At this point, I began using equations like these:
My students agreed that the first equation is false. Some solved for both sides and said that 20 does not equal 19. One student said, “I knew right away that it was false because there is a three on both sides, but 16 is one less than 17.” This student was thinking relationally—this is a critical component to algebraic reasoning! Additionally, the student was using her number sense and looking at relationships among the numbers. She actually did not need to solve for both sides, rather understanding the relationships on each side of the equation helped her figure out that the statement is false. Her explanation of her reasoning helped other students look at the other two problems in the same manner.
Later, after changing the routine from True/False Statements to Open Number Sentences, I had students come up with their own Open Number Sentences that I could use for future Number Sense Routines:
Luis’s Open Number Sentence encourages his peers to use relational thinking rather than solve for both sides. Students who use relational thinking will likely use a compensation strategy, which is a strong mental math strategy.
These routines helped my students negotiate meaning around the equal sign, dispel misconceptions about the equal sign, learn to think relationally, and use important arithmetic and mental math strategies. Through our various conversations each day for several weeks, I watched my students’ number sense flourish. I found these algebra routines to be extremely effective! The students immediately get sucked into the discussions and lose themselves (and find algebraic thinking!) in debate and negotiations about the mathematics. Additionally, a focus on the symbolic (the equations) was a nice addition to our visually focused dot cards and ten-frames routines. At many points in our discussions, I found it helpful to students to link the dot cards with our True/False Statements. This routine is extremely rich and can be adapted and enriched in a number of ways!
We continue our math quick tip series with another idea from Jessica Shumway’s recent book Number Sense Routines. Counting the days in school is a routine that Jessica uses at the end of the day to help separate it from the calendar routine, after she and fellow teachers realized that students were confused about counting the days of the month and the days spent in school at the same time. Read more about this routine here and then head over to the Stenhouse site where you can still preview Jessica’s book in its entirety!
Counting and keeping track of the days in school is an especially beneficial routine for kindergarten and first-grade students. This routine lends itself to talking about numbers, thinking about patterns, and seeing amounts. It provides an opportunity for these young students to count every day, see and experience an increasing amount, and think about numbers beyond 100. For second and third graders, there are a variety of reasons and ways to keep track of the days in school, from organizing a growing amount to developing sophisticated strategies for comparing two sets of numbers (days in school versus days on the calendar).
As a mathematics coach at Bailey’s Elementary, I worked with a team of kindergarten teachers who described a problem they came across with keeping track of the days in school. They realized that students were getting very confused between how many days are in a month and how many dayswe had been in school during that month. “What are we counting?” became a common question. Teachers were not asking students to compare the days in a month versus the number of days students had been in school. The problem was that there were too many different numbers (day of the month and the number of days in school) for them to keep track of, especially early on in the year.
One of the kindergarten teachers and I decided to use this routine of Counting the Days in School at the end of the school day as a way to remedy the confusion. That way, the calendar routines, which students worked on during morning meeting or at the beginning of the math lesson, were separate from the Counting the Days in School routine. We used Counting the Days in School as a check-off system: “We are finishing the ninth day of school. Let’s add 9 to our counting tape and move our circle on the number grid from 8 to 9. Wow, you’ve just finished up another day of kindergarten. You are nine days smarter!”
I have seen many different ways to keep track of the days of school. Many teachers use a place-value pocket chart, with each pocket labeled from left to right as Hundreds, Tens, and Ones. They add a straw to the Ones pocket for each day they are in school. Every tenth day of school, students bundle the straws into a ten and place the bundle in the Tens pocket on the chart.
Although this routine is effective in third-grade classrooms, it does not seem to be very effective for kindergarten and first grade. Students at this age are in the process of constructing early ideas of number sense and are not yet near understanding why you bundle straws every tenth day of school. This routine requires students to have an understanding of unitizing—counting ten straws as one bundle of straws or one ten. Students in kindergarten and first grade are grappling with early ideas of how we count objects and represent the count with symbols. Counting ten objects as “one” is difficult when you are still constructing the early ideas of counting, one-to-one correspondence, cardinality, and hierarchical inclusion. Understanding unitizing is a huge leap.
Many teachers believe that the straws routine for keeping track of the days in school is planting the seed for strong place-value understanding as students move into second and third grade. I used to believe that, too; however, I have seen time and again that these young kindergarten and firstgrade students are more focused on what that quantity means and what it looks like. Using cubes instead of bundling straws seems to be an easier way for students to construct early ideas of unitizing and of the importance and efficiency of ten. Opportunities to see ten ones being connected to one ten (without the exchange that takes place with bundles of straws or base ten blocks) will help these younger students construct the ideas of “ten-ness.”
The idea of ten as a group is at the core of unitizing. Early on, though, many children are learning that 1 means one item. It is too confusing to bring in the idea that 1 can also mean one group of ten. That will come later. It is more important for very young children (kindergarten and first grade) to build visual images of the amounts rather than focus on unitizing. Collecting items (like rocks or cubes) for each day of school and counting by ones seems to be a more authentic and age-appropriate task for students who are still figuring out what twenty looks like, how to count twenty efficiently, and how to represent that number. The place-value chart does not yet make sense. Let’s shift the focus for these young learners and instead create routines that will help them see amounts, learn the counting sequence, construct a sense of quantities, and recognize patterns.
In the first part of our interview with Jessica, she defines number sense, talks about why it’s difficult to assess it, and how a lack of number sense can hinder students in reaching their potential in math. In the second part, Jessica explains how the routines in her book help teachers learn about students’ mathematical thinking over time, and how math talk with peers can encourage reluctant math learners.