Saturday, February 6, 2016

Numberless Problems: An Adventure in Leaping Before Looking

Sometime before Christmas break, I saw something on my Twitter feed that intrigued me, so I asked a Second Grade teacher I work with if I could borrow her class so I can try it out. Numberless word problems. This idea may seem counterintuitive and I think that's the point - or at least that's the point I'm getting right now.

Let me explain.

Word problems have been a staple of the mathematics classroom for years now and students usually struggle with them. So, we teach them tricks like pulling the numbers out of the problem, and using key words to figure out what to do with the numbers - "altogether means plus them," while "how many left means minus them." And oftentimes these tricks work, but do these tricks get at the point of "doing" mathematics?

Part of mathematics is calculation, but I'm finding that the larger part of mathematics (and maybe the whole of it, I don't know) is sense making. The beauty of the idea of numberless word problems is that they get in the way of teaching tricks and force students into the realm of sense making.

The students I worked with before Christmas break didn't seem to feel any disequilibrium created by this new experience and if they did, they didn't let it get in the way of engaging in the task. Our entry point into the task was a "story" that read something like this, "Mrs. Johnson has some jars on the counter with candy in them."

No numbers. No question. Just a context.

After chorally reading the story, I invited the students to quietly reflect on the sentence. "What are you wondering? What would you like to know about this?" After I saw that the majority of the class had at least one idea (I used a procedure in which students placed a thumb next to their hearts to indicate that they had at least one wondering to share, and each subsequent wondering was communicated with an additional finger), I asked the students to turn and talk in groups of 2-3. The turn and talk drew everyone in the class into the conversation and provided students who would not otherwise share their ideas w/ the whole class an opportunity to share their ideas. Because the students were on the carpet and close, I was able to lean in and listen to several of the conversations to gauge the general direction the class was heading in before brining the group back together. This instructional practice provides me with opportunity to gather formative assessment data in other lessons.

Once the group was refocused, I opened the floor for conversation by asking, "Who would like to share what their group discussed?" Student responses (and the information I provided after they asked) included:

  • "What kind of candy is it?" (A wide variety, but mostly caramels because caramel is delicious!)
  • "How many jars are there?" (8)
  • "Who is the candy for?" (For a party Mrs. Johnson is planning)
  • "How much candy is in each jar?" (6 in each)
  • "Do we get any candy?" (Good question! I hope so!)
  • "How many candies will each of us get?" (Hmm... I'm not sure. What would we need to know in order to figure that out?)
Some of the wonderings were what I call "Math-y wonderings" while others were things any living, breathing human would want to know in a situation about candy. One thing that is important to me that my students know about math is that all of their ideas are valuable and welcome. Therefore, I recorded all of the wonderings shared. (Another thing my students soon learn in that all ideas are up for close and critical examination. All ideas are welcome because ideas are interesting. But the group gets to examine those ideas graciously and respectfully to determine their accuracy and/or relevance to the math task at hand).

Before going into the classroom, I was thinking the task would eventually turn into a repeated addition problem and I tried to steer it that way, but they wanted to investigate how much candy each of them would receive. The students took the problem to a higher level. The teacher (me) wanted to reign them in. This is an awesome and occasional byproduct with open mathematics like this.

"What else do we need to know if we're going to determine how many candies each student will get in this class?"

"We'd need to know how many kids are in the class." Students start counting each other.


"No! There's 4 people absent and you missed someone. There's 22."

"Yeah, I see 22."

"Do we include the people who aren't here?"

"It's your party," I responded. "How would you like to make this happen?"

The class decided on leaving those who were absent out of the party.

Our problem now read, "Mrs. Johnson has 8 jars on a shelf. Each jar has 6 candies in them. How many candies will each of the 23 people in the class today get?" (They decided Mrs. Johnson would get candy, but for some reason I didn't get any. It's an oversight, I'm sure!)

The time I had allotted for this experiment was up. I thought we'd just define the problem, but the kids now wanted to answer the question they'd posed to themselves. I asked Mrs. Johnson if I we could keep going and she graciously agreed.

So, I set the kids free to go back to their desks to get after it. Some students drew pictures, others wanted to use snap-cubes, while still others made calculations. While students grappled with finding a solution, I monitored student progress and use of various strategies and began mentally selecting student work samples to use during our debrief.

For the most part the productive struggle in the class was evident. Some students plowed ahead on their own, having a clear idea of what they were doing. Others turned to their table groups and asked clarifying questions of each other. There was a handful of students toward the front of the classroom who only said, "I don't get it." Luckily they were grouped close to each other, which made it easy to scaffold the problem with them.

"Okay. So, you 'don't get it.' What do you get? Let's read through this together and see what you do understand."

Often I find that students who are historically "bad" at math have found that if they just sit long enough, someone will essentially do the work for them. By coming along side of them and using questions to productively probe their understanding I communicate that I believe they CAN do the work while at the same time assist them in making meaning of the situation for themselves. I work hard at not contributing to the curse of learned helplessness that many students have developed, but it's not always easy.

As we worked our way through the problem and identified what they did understand, we drew rough sketches of what they knew. We talked about what was happening to the candies and how we can represent that in "math language" through mathematical notation. I left the students to continue their work and they made significant progress toward finding the total candies. One of the great things about an open problem like this is that there are entry points for every learner and every mathematician is working on math they need to develop.

Before bringing the kids back together, I continued monitoring and selecting student work samples, while at the same time mentally sequencing them. For debriefing a problem, I like to sequence samples from the most concrete to the most abstract. By ordering the samples in this way, students who, "Don't get it" will most likely have something to grab ahold of during the debrief - they can see the solution in ways that are most basic and representative of the problem.

Once we begin debriefing I work on helping the students make connections between the sample and the problem by asking students to explain why their classmate did what they did. We also spend time comparing and contrasting strategies. These layers of questions drive students to make meaning of the math - to help them see that mathematics has little to do with calculations, but with understanding and overcoming obstacles.

I ran out of time with this class. We got most of the way through our debriefing and the kids had to go to music and I had a meeting to get to. But, if it were my class, I might have stopped the lesson before the debrief in order to let their work simmer for a bit. They'd worked hard for an extended period of time and I want them to be mentally fresh during the debrief. I'd also like those few students who were not quite done to persevere and finish and have that reward of completing something that was challenging.

So, that's my first attempt at a numberless word problem. Not sure I did it "right," but what I saw students doing, the engagement, the level of productive struggle, and reasoning made me think what we did was worthwhile.

If you'd like to read more about numberless world problems, check out Brian Bushart's posts here and here. If you'd like to learn more about facilitating productive conversations around student work samples, checkout NCTM's 5 Practices for Orchestrating Productive Mathematics Discussions


  1. Thank you for sharing the work you and your colleague and her students did. The combination of numberless word problems and the 5 Practices is so powerful. Am going to use your blog post as a way to open the conversation with the grade 2 team in our work session this week. Appreciate your generosity.

  2. Great! I'm curious to know how it went!

  3. Great work, Chris! I like that you left the question off for them too. As a first grade teacher I employed numberless word problems to help me teach compare problems (amongst other things). They are so tricky that it is non-negotiable to slow down and pay close attention to the words. I think you did a really nice job giving struggling students access to the math as well as implementing the 5 Practices. I always like to add a 5th where we reflect after the connect. I'm looking to see if kids progressed through CRA and efficiency. Thanks for sharing!

    1. Thanks Jamie! Those compare problems! Ugh! I'd love to hear more about the 6th practice and how you use that time to see how the students are progressing through CRA. One thing I did in a 4th grade classroom this year was to have them journal between students sharing their strategies. My prompts were 1) How is student x's strategy similar/different from student y? 2) Which strategy would you like to try in the future and why? Lot's of interesting responses. Thanks again!