Yesterday was an exciting day in first grade: the launch of peas & carrots! This is the first “how many of each?” problem in first grade. Students wrestle with two missing addends to create a target sum.
In the launch of the problem, a teacher’s job is to ensure that students understand the context enough to enter into the problem. With a solid launch, the teacher is then able to spend more time observing students, and asking questions. First grade teacher Rachel launched Peas & Carrots with a story.
I was making dinner last night. You know how I love vegetables, so I decided to get out a big pot and cook up some peas and carrots. I stirred them.
“Oh, like a salad!” Marco shouted out.
Well, these vegetables were cooked, Rachel continued. I got out a ladle. Do you know what a ladle is? A giant spoon. I got out my ladle, and I spooned out 7 vegetables onto my plate.
Rachel drew a large circle on the board.
“Ohhhh, it’s not a bowl, it’s a plate!” Marco interjected.
Rachel elicited one possible combination from the class, and then sent the students off to work. She asked me to work with two students who frequently benefit from some extra processing in order to feel ownership over a problem-solving context.
I asked the two girls — Charlotte and Gracie — to tell me the story again. Neither felt confident about the details. “Was Ms. O’Sullivan making a salad?”
After repeating the story and hammering out the question — how many peas might there be? How many carrots? — Charlotte set off to work.
Meanwhile, Gracie was spinning her wheels. She drew a rough sketch of a person with long hair with a thought bubble.
“This is how I show my thinking!” She announced proudly.
“Okay. So what are you thinking about this problem?” We repeated it again.
“There are 7…” She trailed off.
“7 peas and 7 carrots.”
“Hmm… I think there are 7 vegetables in all.”
“7 peas… 7 carrots…”
“Would that make 7 in all? There are 7 vegetables in all. So what if there were 4 carrots?”
Gracie drew 7 circles, and then 7 more circles.
“There are 7 peas and 7 carrots!”
“Is that 7 in all? That looks like more vegetables than just 7.”
“So there are 7 peas and 7 carrots and then I eat 7 and now I have 7.”
“That feels like a different story than the one Ms. O’Sullivan told. There are only 7 vegetables in all. Here, let’s say that these green unifix cubes are peas, and these orange unifix cubes are carrots. We have only 7 cubes — vegetables — in all. So…”
“7 and 7 and then I take away 7 and now I have 7. Is that 7 + 7?”
It was like Gracie and I were in parallel universes. We were not engaged in the same problem. I paused so I could consider how to help Gracie enter into our “how many of each?” situation.
Charlotte sighed. She was recording her second solution at this point.
“There’s no eating! There’s just ‘had.’” Charlotte nudged. Gracie blinked.
“So take away the 7…”
“No, that’s eating! There’s no eating. Only had.”
“Ohhh,” Gracie murmured. I’ve seen this move before: students, when they feel like they should understand something that still feels confusing, offer up a melodic “ohhhhhh” that is basically a polite “please leave me alone.”
But that wasn’t actually what Gracie was communicating. What Charlotte had told her was actually a breakthrough: there was no action to take. We were just mathematizing what was there, and there are different possible combinations.
“If I want something to equal 7,” Gracie started. “I know that 6 and 1 more make 7.”
“That’s true!” I affirmed. We had been working on counting on and back earlier this week to make the connection to addition and subtraction. Maybe it was working…
Gracie drew 6 carrots and 1 pea on a plate, with a line dividing the two sides. Another student had suggested this in the launch of the problem.
When finished, Gracie triumphantly held up her book to show me her work.
“Wow! 6 carrots and 1 pea make 7 vegetables in all. 6 plus 1 equals 7! That works!”
I looked over at Charlotte’s work. She had carefully written 4 + 3 = 7 and 3 + 4 = 7.
“Wow, it looks like Charlotte has found another answer that works. 4 + 3 = 7. What do you think about that?”
The girls nodded, but didn’t say much. “Yeah!” So I continued: “do you think there are even more ways to do this?”
“Gracie picked up her pencil quickly, which is honestly not something I’ve seen her do very often. Most days, she moves at 0.5 speed through the classroom. She winds her way around the table groups to get anywhere that she’s going.
“1 plus 6!” She exclaimed.
“1 plus 6?”
“1 carrot and 6 peas!” Gracie clarified. “It’s kind of the opposite but it’s really the same.”
We discussed what felt like it was the same and what felt different (or the ‘opposite’). Later, during the class discussion about his problem, Gracie eagerly volunteered to share her work (and Rachel chose her to share).
This is all to say that some of the most important work we do is to help students understand the task without necessarily telling them how to approach this task. This is critical to problem-based learning.
“There’s no eating. There’s just ‘had’!”
There’s a good chance that, if I had used this brilliant words, Gracie would have had a breakthrough all the same. There is also power in the fact that these words came from 6-year-old Charlotte, Gracie’s friend. It positioned Charlotte’s ideas as valuable, and that there is learning we all do with and from one another.
There’s a fairly well-known article called “Never Say Anything a Kid Can Say,” by Steven C. Reinhart (published in NCTM’s Mathematics Teaching in the Middle School: April 2000, p. 54 – 57). This phrase often comes into my head as I’m working in a classroom, trying to center the learners. The article is always worth revisiting, too.
We can’t always plan for these moments, but we can celebrate when they happen.