Recently, I watched an NCTM shadowcon talk by Robert Kaplinsky. I can’t stop thinking about it. His words echo in my thoughts. I am a district math coach. I have no administrative power. I only have the opportunity to influence. Sometimes, my influence is positive. Sometimes my influence is negative. Often, the difference between the two is how honest I am with myself about my intentions and how intentionally I reflect.
Last week, I led a learning round in an elementary school. I watched an exceptional lesson in a third grade classroom. What made the lesson exceptional was how little the teacher said. For ten minutes, 7 different students participated in a student generated investigation about how many lines you need to draw to show fourths on a number line.
When we walked in, we heard:
Student #1: “You have to draw more than three lines because if you only drew three lines, you would have thirds.”
Student 2: “I agree. If you didn’t have the end line, you would have thirds.”
Student 3: “You don’t count all the lines. You don’t count the 1.”
Student 4: “I see both sides of the story. What does the zero stand for?”
Pause. A long, silent pause.
Student 5: “If you count the 1 line, you have to count the zero line.”
Student 6: “I disagree with you… about not counting the 1 line. If you didn’t count the 1 line, you would only have three fourths. You would only have three pieces. It really wouldn’t make sense without the end line.”
Student 7: “Yeah! If you don’t have the zero line and the 1 line, the numbers would go on forever. The zero and the one are like the start and the stop.”
Several Students: “I agree with (student 7). A number line goes on forever. When you make fractions on a number line, it is kind of like you are showing the pieces. You need the zero line and the 1 line to show the “piece ” of the whole number line.”
Another long, silent, pause.
Mrs.Watkins: “I think I heard you say a couple of things. When showing fractions on a number line, it is really like showing a piece of the whole number line – a line segment. We need to draw the zero line because it tells us where to start. We need to draw the 4/4 or 1 whole line because it tells us where the whole ends. You taught me something today. I could be more specific when I am using number lines to show fractions. I could call the fraction pieces line segments.”
These students presented, questioned, and defended their own and each other’s reasoning. The teacher listened.
I was participating in this learning rounds with an interventionist from another building. She didn’t know these students at all. Afterwards, I asked her, “Would you be able to guess which students received “gifted and talented” services?” She said she would have no idea. Then, I asked her, “Would you be able to guess which students receive interventions?” She said she would have no idea. I pushed. I asked her, based on the thinking she just observed, choose a student who you think sounded like a typical “gifted and talented” student. She chose a student who receives math interventions.
After I left this classroom, I couldn’t stop thinking about these students and their teacher. Their words were echoing in my thoughts.
During the debriefing session of our learning rounds, I asked my peers who observed after we did, “What happened next?” I was hooked. I only saw ten minutes of this math lesson. What else did these students do and say after I left?
They shared with us that, after we left, the teacher presented the students with a problem about cupcakes. “There are 3 cupcakes and 4 kids. How should they share? I wonder if they should each get half?” She sent the students off to work in small groups to come up with a solution.
As the observing teachers circulated, they noticed the level of engagement in math talk. One student said to her partner, “no offense to Mrs. Watkins, but I think her estimate is off. I am pretty sure each kid could get more than half a cupcake.”
Later that day, I checked in with Mrs. Watkins. I asked her what the kids came up with for solutions. She showed me the white board below and said, “this was really interesting.”
“Is carot a type of cupcake?”, I asked.
“No. It is the name of the kid. They chose their own names.”
Of course. Carot, Tomato, Charlie, and Joe. Just your typical group of nine year old names.
She told me that these students presented to the class that the solution could either be 3/4 or 1/4 or 3/12. She endured yet another silent pause as I processed.
“1/4?”, I asked. She was being really patient with me.
“1/4 or 3/12 of all three cupcakes. These students explained to us that the answer depended on what you considered to be the whole.”
Again, she waited for me to catch up. I felt humbled. I wouldn’t have thought to consider either of those answers. She might not have thought of those answers herself, but she was open and quiet enough to allow her students to consider them.
Yesterday, I went back into third grade. I can’t stay away. Mrs. Watkins has inspired me. She’s got me thinking more deeply about fractions. I shared with Mrs. Watkins that, since I left her, I saw this message on twitter:
She and I had a great conversation about how and why we label fractions on a number line. She told me that one of the first lessons she does with her students is create fraction strips. She has the students label each interval as a unit fraction. She doesn’t introduce labeling the hash marks on a number line until later. The lesson that I described above was her first introduction of labeling the hash marks. Again, I learned from her – the progression matters.
Mrs. Watkins has been teaching third grade for 30 years. She often talks to me about how much her math instruction has changed since she first started teaching. When I first started teaching, I worked across the hall from her. She is quiet. I think she would consider herself an introvert. This year was the first year we completed learning rounds in her building. Learning rounds are opportunities for teachers in a building to observe short segments of each other teaching with the intention of learning together to improve math instruction. Some of the teachers that Mrs. Watkins works with said, “We have worked together for 16 years and today was the first time I saw her teach.” I am so grateful that we are all able to see Mrs. Watkins teach. Thank you, Mrs. Watkins.