As a math instructional coach and supervisor for West Windsor–Plainsboro Schools in Princeton Junction, New Jersey, over the past 15 years, I supported teachers in training their students to become good mathematicians. When we began to implement the Common Core, we took a team approach—attending workshops together and meeting up for building-based book groups. Along the way, our team developed a few “plays” that helped us put the standards into action and that can help you, too, no matter what grade you teach.
Play #1: Ground Every Lesson in the Practice Standards
When I heard about the Common Core State Standards, my first reaction was probably similar to yours: What do I have to teach now? Resigned to the inevitable, I read the content standards, which outline the expectations at each grade level. But it was almost a year before I went back and read the eight practice standards that are tucked into the first few pages of the document. When I did, I immediately realized why they appeared first: These are the daily practices that we should infuse into every math lesson.
The practice standards are the processes and traits of good mathematicians, no matter their age. A first grader or Eratosthenes, a junior in high school or Einstein, all should aspire to these practices.
Play #2: Build Perseverance in Problem Solving
In West Windsor–Plainsboro SD, I worked with a range of K–5 teachers, from those teaching basic-skills math to ones leading gifted and talented classes. But that first practice standard—make sense of problems and persevere in solving them—is universal. Faye Airey, a basic-skills teacher at Millstone River School in Plainsboro Township, agrees: “A lot of my kids give up quickly. I have to help them learn different strategies to approach a problem.” Airey also teaches students to stick with difficult problems. “Perseverance has to be built with small successes,” she says. “We start with simple problems and gradually get more complex. The kids experience struggle, but it is a supported struggle.”
Play #3: Emphasize That Math Is More Than Being a Calculator
Renee McFall, a third-grade teacher at Town Center Elementary School in Plainsboro, says most of her students think that good mathematicians merely churn out accurate computations. McFall uses projects to help kids make real-work connections. Her class recently created and sold yarn bracelets, with all proceeds donated to a local charity. The students bought supplies, created budgets, drew up a plan, and convinced “investors” (teachers) that the product would make money. They needed to be precise in their language and calculations, measure accurately, and have a logical plan—because mistakes cost money.
Play #4: Have Students Model the Math
Many students learn rote math in a rote way. They know how to do a certain process, but they don’t understand how or why it works. To develop a deeper understanding, I used a simple technique I call “draw the line.” I gave students a problem to solve and then asked them to draw a line beneath their solution. Underneath that line, I asked them to solve the problem another way. They might draw a picture, construct a diagram, or build a physical model that proves their equation is true. For example, if a student writes 4 x ½ = 2, look for some kind of model that shows what is happening and proves that the equation they wrote works.
Play #5: Teach Active Explaining and Listening
When I modeled lessons in classrooms, I often told students a story about why explaining your thinking is important. Take, for example, an engineer who designs a bridge for a community. The planning board asks him, “Can you explain how you came up with this design and why we should choose it?” He says, “No, I just know it’ll work.” Do you think he’ll get the contract to build the bridge? The third practice standard—construct viable arguments and critique the reasoning of others—is a skill we can develop only by giving students opportunities to explain their ideas.
“Sharing multiple solution methods is important for kids to become better problem solvers,” says Shanna Weber, a gifted and talented teacher at Millstone River School. “They have to explain their solutions clearly when sharing. That’s a skill. And so is actively listening to your peers’ explanations.”
“Echo” is a good technique to promote this kind of listening. After a student shares a strategy, other students “echo” back that idea in their own words. Suddenly, students need to attend carefully to classmates’ ideas. This supports engagement, and with that comes more learning.
I see these practice standards as essential tools to help kids develop as great mathematicians and learners at large. Airey said it best: “Nobody’s asking, ‘When are we going to use this?’ They see it makes sense. We model and use these practices every day in meaningful ways. They are skills and processes students can use for life.”
Ready to put the standards into action? Try the following activities, broken down by grade band.
All About My Number
(Standard 4: Model with mathematics)
Assign each student a number from 1 to 20. Have students write the number on a sheet of poster paper. Then, ask them to show their number every way they can imagine on the poster. This could include word form, equations using addition or subtraction, tally marks, drawings of objects, number lines, and anything else that makes sense. Have students share their finished pieces so they can see the many different ways their classmates represented numbers.
Are You Wearing a Pattern?
(Standard 7: Look for and make use of structure)
Pair students. Have them look carefully at the clothes their partner is wearing. Ask if they can find a pattern anywhere in the clothes. It might be stripes or checks or buttons or colors. Invite students to discuss why these are patterns (they repeat) and then record them using crayons and paper. Share patterns and discuss.
Math Word Wall
(Standard 6: Attend to precision)
A math word wall can help students become precise with the terms they use. As you encounter new math terms, assign students to take turns creating vocabulary cards for the wall. Each card should include the word, a definition, and a diagram or example. These will serve as great reference tools for future lessons.
What Do You Need?
(Standard 5: Use appropriate tools strategically)
Instead of putting out materials for students to use with an upcoming project or problem, challenge them to generate a list of tools they’d like to use and why. The list might include graph paper, blocks, or a ruler. This exercise sounds simple, but it requires students to brainstorm problem-solving strategies. It also helps them focus on the role of tools and materials in mathematics.
The Diamond Ring
(Standard 1: Make sense of problems and persevere in solving them)
With word problems, I typically used five steps that correspond to the five fingers on a hand: Identify the question, identify the facts, eliminate what you don’t need, choose a strategy and solve, and check that your answer makes sense. Recently, I added another step, one I call “the diamond ring.” Challenge students to identify the question and write an answer sentence with a blank. For example, with the question How old is Mr. Krech?, the answer sentence would be Mr. Krech is ___ years old. This activity helps students figure out and focus on exactly what they are looking for.
Letter to an Alien
(Standard 3: Construct viable arguments and critique the reasoning of others)
Ask students to solve a word problem and then explain their thinking by writing a letter to Bizbop the Alien. Students should assume Bizbop knows very little math, so they should detail all of their computation and thinking, including what they did and why they did it. I have used writing to aliens with larger questions, such as “Can you explain our system of liquid measurement to Bizbop?”
(Standard 8: Look for and express regularity in repeated reasoning)
Have students use pattern blocks to make a train of squares. Ask them to count the number of squares and calculate the resulting perimeter in units (a unit being the length of one side). Have them use the squares to figure this for up to 10 squares. Ask if they notice any patterns. If so, can the pattern be expressed in a rule or algebraic formula? Then, ask them to use a larger number like 19 and plug it into the formula. Test with a square to see if it works.
Here’s the Answer. What’s the Question?
(Standard 2: Reason abstractly and quantitatively)
Say, “I have an equation I’ve already solved. Write a word problem that would fit with it.” If I write (2 x 25) + 10 = 60, a student might respond with this: Jill worked for two days painting a fence for her uncle. He paid her $25 a day. She did such a good job he tipped her $10. How much money did she earn?
A recipient of the Presidential Award for Excellence in Mathematics and Science Teaching, Bob Krech has worked as a teacher, supervisor, administrator, and curriculum specialist.
Image: Dan Saelinger