Students' Ideas at the Center Part 2: Liz Romero's Math Class
Part 1 of this series examined knowledge (Knowledge is Connected). What could this look like in practice? Join me in Liz Romero’s 3rd Grade classroom.
This post focuses on in-person instruction, but many of the ideas translate to online instruction. For tips on remote teaching, check out this post.
It’s November, so students have internalized the routine of math class: 5-10 minutes warm-up and 40-45 minutes problem solving.
Today the warm-up was a true/false equation. Liz wrote “4×5=5+5+5” on the board, gave students 20 seconds of silent think time (they put a thumbs up by their chest when they had an idea to share), then gave them a minute to talk to a partner. Liz listened in and strategically called on a few students to share. She asked each student a follow-up question that increased their precision. If a student said “I added the 5 and 5,” she said “which 5?” If they said “I know 4×5 is 20,” she said “How did you know?”
“Warm-ups are an opportunity for me to build on student ideas, invite curiosity for students to make connections with numbers, and to inspire students to see relationships in their discoveries.” – Liz Romero
Next is Problem Solving, where the teacher poses a problem for students to solve without any instruction. Liz uses contexts that are meaningful to her students. In this case, she is actually bringing cookies to class tomorrow! Students discuss the context but not the math.
The number sets give students choice while engaging everyone in the same mathematical ideas. The fourth set (?, ??) allows students to choose any 1 digit by 2 digit multiplication. Students work independently for 10 minutes in their math journal in black pen. Why not pencil?
“I have students use a black pen because I want to see everything that they are thinking, I do not want them to erase anything. Seeing all of their ideas/strategies and how they are thinking about the number story helps me better understand and support them as mathematicians.” – Liz Romero
Then, Liz gives the class a one word cue: “Pens.” Students quickly grab a different color pen, choose a partner, and work together on the problem.
“I have students use colored pens [after they finish their independent work] because it helps me distinguish what they did on their own and what they learned from their partner(s). With new learning represented by the colored ink, I am able to see how talking about math with their partner(s) could help them; revise their thinking, see new ideas and connections, push their thinking to try their friend’s strategies or push their thinking to try other number sets.” – Liz Romero
Side note: Liz prints labels with the problem so students don’t have to copy the problem by hand.
The class enjoys the time to collaborate as they share ideas and correct themselves (no need for the teacher to be the “answer key”). Liz confers with students and uses multiple follow-up questions to gather formative data. By the end of this part of the lesson she has detailed mental notes on about 10 students.
Liz chose this problem type (multiplication) and numbers (6 groups and 12 in each) to surface student strategies around the distributive property. This is her secret, however, since the goal of Problem Solving (from the students’ perspective) is to make sense of the problem and solve it in any way that makes sense. You’re not problem solving if the teacher shows you how to solve!
“I selected the number choices because we had been working with groups of 10. I made sure to include one number in each set that was close to 10. I chose 11 and 12 to see if they would decompose the numbers to help solve the number story. If you notice, I also switched one of the number sets to see if anyone caught The Commutative Property creeping in.” – Liz Romero
During the share-out, students recreate their strategy on a new sheet of paper, which allows the class to process the explanation both verbally and visually. Next to the document camera is a cup of colorful sharpies, used to color code ideas and make the strategies clearly visible when projected.
Liz purposefully picks two students to share aloud how they solved. When compared and connected, these two strategies reveal important ideas about the distributive property. Can you predict what might come up?
The first student solved the problem by drawing 6 boxes with 12 cookies in each. To save time, Liz had this student draw all but the last box before the share-out started. The student explains their strategy all the way through and answers follow-up questions from their peers.
One of the most surprising things about Liz’s class is their collective autonomy. It takes a lot of effort to step back and let kids take the lead. How did she do it?
“When we first started having conversations about math, I made sure that the math we were talking about was ‘easy.’ An example of this could be, ‘Our class has two boxes of pens. One box has six pens. The other box has five pens. How many pens do we have?’ Since all the third graders in my class could solve the number story easily, the task was not about the math. This was my opportunity to practice having conversations of how students solved and how we would respond to each other. I also initially used sentence stems, speech bubbles glued to sticks and walked back and forth between the students that decided to chime in. We practiced having discussions a lot. Eventually, the conversations became more organic throughout the year.” – Liz Romero
If you walked into Liz’s class during a discussion you might wonder, “Where’s Liz?” To encourage students to engage with each other (instead of speaking to each other through the teacher) Liz sits at the back of the carpet and redirects any questions or comments back to the class. “Don’t ask me, ask him!” “I don’t know, what do you all think?”
You might be wondering: What if they say something wrong? When the teacher isn’t the “answer key,” a collective responsibility becomes the norm. In a beautiful manifestation of positive assumption, a student asks: “Did you mean to put 11 in that one?” “Oops, that was supposed to be 12.”
Also, What if they don’t ask the right questions? Students often know what they’re confused or curious about, so why not cut out the go-between (the teacher)? Students who lean into confusion and curiosity are learning how to confidently navigate the act of learning. Liz gives her students a chance to practice this every day, and today they don’t disappoint: “Why did you draw all the boxes? Was that more efficient for you?” “How did you know it would work to break up the 12?” “I’m confused where the 10 came from, can you explain that again?”
Okay, back to Liz’s secret plan to discuss the distributive property. The next student drew 6 boxes but wrote the number 12 inside of each instead of drawing each cookie. Then, the student wrote 12+12+12+12+12+12, decomposed each into 10 and 2, added the 6 tens to make 60, and 6 twos to make 12. Finally, they added 60+12 to arrive at 72. The class asked follow-up questions to dig into the strategy.
Liz thanks both students, positions the strategies next to each other under the document camera and asks an open-ended question: “What is the same and different about these strategies?” She gives them silent think time, then has them discuss with a partner. Students notice 6 boxes of cookies represented in each, the cookies individually represented in the first strategy, and the equation in the second. Liz knew most of her class wasn’t decomposing teen numbers when multiplying, so she focused in on that: “Does the 10 and 2 from the second strategy appear in the first strategy?” (Long pause) “Where?” She has multiple students explain their thinking, annotating on the board to show how decomposing 6 twelves into 10 and 2 is the same as drawing out 6 boxes with 12 cookies in each.
After a substantive conversation, students are ready to apply new ideas or revise their work. Liz always gives students 5 minutes to apply what they’ve learned: “You can revise your work or try a new number set.” She takes the pieces of student work and staples them onto the math wall – a ongoing celebration of student work that says: “We use our own strategies to solve math problems.” This acts as a celebration of student thinking and a way for Liz to track who has shared more (thicker stack) and less so she can lift up every mathematician in her class over time. With the last few minutes she checks on a handful of students who were stuck earlier.
As I leave Liz’s class, a few things are on my mind.
First, how can the structure of the classroom and the activities we choose bolster student agency? Everything Liz did reinforced the idea that her students were capable of collaborating with each other to solve and discuss the math. Her presence was to facilitate the process, and she constantly redirected attention away from herself. Here is a list of choices we can give students during math class from Tracy Zager. I highlighted the choices Liz gave to her students just from today.
Second, context is the entry point and foundation for abstract mathematical ideas. The students discussed a hefty math concept (distributive property) with ease – all because the context brought the numbers and symbols to life. The 12 being decomposed into 10 and 2 made sense, because of course you could count 10 cookies from each box and have 2 more cookies from each box to count. Today’s experience is a single thread in a larger tapestry of number sense, slowly woven by a crafty and patient facilitator. Liz is playing the long game.
Finally, we’re so thoughtful and reflective that we sometimes overlook our strengths. Liz told me at least 5 things she wanted to do differently in this lesson alone. If you’re like me or Liz, it is easy to focus on what to improve. Take a moment and think about all the wonderful things I would notice if I walked into your classroom.
Thank you Liz for your incredible dedication to your students and for sharing your practice with us. Follow Liz on Twitter here.
Questions to consider:
- What are you already doing to put student thinking at the center of math class?
- What surprised you about Liz’s class?
- What might you try or tweak for your own class?