Purposeful Numberless Word Problems

[UPDATE – You can find all of my numberless word problem sets on this page.]

This year I read Sherry Parrish’s Number Talks from cover to cover as I prepared to deliver introductory PD sessions to K-2 and 3-5 teachers in November. She outlines five key components of number talks; you can read about them here. One of the components in particular came to the forefront of my thinking the past few days: purposeful computation problems. I’ll get back to that in a moment.

It all started when I got an email the other day asking whether I have a bank of numberless word problems I could share with a teacher. Sadly, I don’t have a bank to share, but it immediately got me thinking of putting one together. That led to me wondering what such a bank would look like: How would it be organized? By grade level? By problem type? By operation?

That brought to mind a resource I used last year when developing an extended PD program for our district interventionists: the Institute of Education Sciences practice guide Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. The guide lays out 8 recommendations. I was reminded of this one:

Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures.

Students who have difficulties in mathematics typically experience severe difficulties in solving word problems related to the mathematics concepts and operations they are learning. This is a major impediment for future success in any math-related discipline.

Based on the importance of building proficiency and the convergent findings from a body of high-quality research, the panel recommends that interventions include systematic explicit instruction on solving word problems, using the problems’ underlying structure. Simple problems give meaning to mathematical operations such as subtraction or multiplication. When students are taught the underlying structure of a word problem, they not only have greater success in problem solving but can also gain insight into the deeper mathematical ideas in word problems.

(You can read the full recommendation here.)

And it was this recommendation that ultimately reminded me of the part of Sherry Parrish’s book where she talked about purposeful computation problems:

“Crafting problems that guide students to focus on mathematical relationships is an essential part of number talks that is used to build mathematical understanding and knowledge…a mixture of random problems…do not lend themselves to a common strategy. [They] may be used as practice for mental computation, but [they] do not initiate a common focus for a number talk discussion.”

All of this shaped my thoughts on how I should proceed if I were to create a bank of numberless word problems to share. Don’t get me wrong, the numberless word problem routine can be used at any time with any problem as needed. However, the purpose is to provide scaffolding, and we should provide scaffolding with a clear instructional end goal in mind. We’re not building ladders to nowhere!

The end goal, as I see it, is that we’re trying to support students so they can identify for themselves the structure of the problems they’re solving so they can successfully choose the operation or operations they need to use to determine the correct answer.

In order to reach that goal, we need to be very intentional in our work, in our selection of problems to pose to students. We need to differentiate practice for solving problems from purposefully selecting problems that initiate a common focus for problem solving.

What Sherry Parrish does to achieve this goal with regards to number talks is she creates problem strings and groups them by anticipated computation strategy. I didn’t create problem strings, per se, but what I did do was create small banks of word problems that all fit into the same problem type category. I’m utilizing the problem types shared in Children’s Mathematics: Cognitively Guided Instruction.

Here’s the document the image came from. It’s a quick read if you’re new to Cognitively Guided Instruction or if you want a quick brush up.

So far I’ve put together sets of 10 problems for all of the problem types related to joining situations. I plugged in numbers for the problems, but you can just as easily change them for your students. I did try to always select numbers that were as realistic as possible for the situation.

• Join – Result Unknown
• Join – Change Unknown
• Join – Start Unknown

My goal is to make problem sets for all of the CGI problem types to help get teachers started if they want to do some focused work on helping students build understanding of the underlying structure of word problems.

I created these problems using the sample contexts provided by Howard County Public Schools. They’re simple, but what I like is that they help illustrate the operations in a wide variety of contexts. Addition can be found in situations about mice, insects, the dentist, the ocean, penguins, and space, to name a few.

As you read through the problems from a given problem type, it might seem blatantly obvious how all of the problems are related, but young students don’t always attend to the same features that adults do. Without sufficient experience, they may not realize what aspects of a problem make addition the operation of choice. We need to give them repeated, intentional opportunities to look for and make use of structure (SMP7).

Even though I’m creating sets of 10 problems for each problem type, I’m not recommending that a teacher should pick a problem type and run through all 10 problems in one go. I might only do 3-4 of the problems over a few days and then switch to a new problem type and do 3-4 of that problem type for a few days.

After students have worked on at least 2 problem types, then I would stop and do an activity that checks to see if students are beginning to be able to identify and differentiate the structure of the problems. Maybe give them three problems, 2 from one problem type and 1 from another. Ask, “Which two problems are of the same type?” or “Which one doesn’t belong?” The idea being that teachers should alternate between focused work on a particular problem type and opportunities for students to consolidate their understanding among multiple problem types.

On each slide in the problem banks, I suggest questions that the teacher could ask to help students make sense of the situation and the underlying structure. The rich discussion the class is able to have with the reveal of each new slide is just as essential as the slow reveal of information.

You may not need to ask all the questions on each slide. Also, you might come up with some of your own questions based on the discussion going on in your class. Do what makes sense to you and your goals for your students. I just wanted to provide some examples in case a teacher wasn’t quite sure how to facilitate a discussion of each slide for a given problem.

Creating these problem sets has prompted me to make a page on my blog dedicated to numberless word problems. You can find that here. I’ll post new problem sets there as they’re created. My current goal is to focus on creating problem sets for all of the CGI problem types. When that is complete, then I’d like to come back and tackle multi-step problems which are really just combinations of one or more problem types. After that I might tackle problems that incorporate irrelevant information provided in the problem itself or provided in a graph or table.

I’ve got quite a lot of work cut out for me!

Writing Numberless Word Problems

So you came across my post on numberless word problems, you got excited by the idea, but you’re left wondering, “Where does he get the problems from?” Good question! I thought it was high time I answer it.

For starters, I try to avoid writing problems from scratch whenever possible. I can do it, and I have done it on numerous occasions, but I’ll be honest, it’s mentally exhausting if you have to write more than one or two problems in one sitting! It takes a lot of work to think of context after context for a variety of math topics, especially if you don’t want to feel like you’re reusing the same context over and over again.

I’ll let you in on a secret. More often than not, I base my questions on existing questions out in the world. I don’t reuse them wholesale, partly because I don’t want to infringe on copyright and partly because I don’t want to deprive teachers in my district of an existing problem they could be using with their students.

I always change names and numbers, and as needed I tweak the contexts and questions. This is so much easier than writing problems from scratch! Basing my problems on existing problems makes me feel like I’m starting 10-20 steps ahead of where I would have otherwise!

I’ll share a few problems I’ve created to give you an idea of what I’m talking about. I based all three of them off grade 3 2015 STAAR sample questions released by the Texas Education Agency.

Problem 1

Here’s the original problem:

 

First, I thought about how I could adjust the problem to make it my own:

• I decided to change the character to Jenise.
• I changed “flowers” to “carrot plants.”
• I changed 21 to 24. I did this intentionally because 24 has so many factors. You’ll see how this plays out when you get to the sample questions I created later.
• I removed the number 3 altogether. Again, this plays out later when I created questions about the situation.

Note: This step is only necessary if you want to create a unique problem. The released tests are free to be used, so you could just as easily convert this exact problem into a numberless word problem. Again, I don’t want to steal resources from my teachers so I’m opting to change this into a new problem.

Next, I think about how I want to scaffold presenting the information in the problem. I create slides, one for each phase of revealing information. Remember, the purpose of a numberless word problem is to give students an opportunity to collaboratively identify and make sense of mathematical relationships in a situation before being presented with a question. There are several factors that dictate how much or how little new information to present on each slide:

• Students’ attention span
• Students’ familiarity with the type of situation being presented
• Students’ familiarity with the math concepts involved in the situation

Here’s how I broke down this question into 4 slides:

Slide 1

Slide 2

Slide 3

Slide 4

Thinking this would be used in a 3rd grade classroom, I opted to break it down quite a bit to draw emphasis on the language of “rows” and “same number in each row.” If I already knew my students were comfortable connecting this language to multiplication and division, then I probably would have combined slides 2 and 3 into one slide.

At this point, I stop and think about what question I want to ask about the full situation on slide 4. If I were a teacher, I might select a question and keep it in my pocket. After discussing slide 4, I’d ask my students what questions they think could be asked about this situation. Students need opportunities to generate problems for themselves, not just be told the problems we expect them to solve. I could allow them to answer their own question before answering the one I had planned (or instead of!).

Here are a few questions I generated that I might ask about this situation:

This is where changing 21 to 24 in the problem adds some richness to the potential questions I could ask about this situation. This is also the reason I removed the number 3 from the original problem. Not specifying the number of rows allowed me more flexibility to ask about either the number of rows or the number of plants in each row.

Problem 2

Here’s the original problem:

I like this problem, so I didn’t want to change it too much. Here are the changes I decided to make. Remember, I always change names and numbers; context and question are tweaked as necessary.

• I changed the character to Mrs. Prentice.
• I changed the food from “yoghurt cups” to “pints of ice cream.”
• I changed the flavors to chocolate, strawberry, and vanilla.
• I changed all three numbers. However, I noted that there was a way to make ten (6 + 4) in the ones, tens, and hundreds places across the 3 numbers, so I tried to create a similar structure in my numbers with 3 + 7.

With those changes, here’s how I scaffolded the problem across 5 slides:

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Depending on my students, I might have combined slides 4 and 5. Keeping them separate means I can play it safe. I can reveal each number one at a time, but I can also breeze through slides 3 and 4 if the situation warrants it and spend more time talking about all three numbers on slide 5.

And finally, it’s time to think of some potential questions that can be asked about this situation:

By the way, this is a great time to point out that I don’t have to pick just one! I spent valuable time crafting the situation and my students will spend valuable time making sense of the situation. Milk it for all it’s worth!

I could pose one question today for students to solve and discuss. Tomorrow we could revisit the same situation, maybe just talking about slide 5 together to jog our memories, and then I could give them another question to solve about this situation. I could even pose 2-3 questions and let the students choose which one they want to solve. Be creative!

Problem 3

Here’s the original problem:

I like this one because I’m able to take a 3rd grade problem and make it fit concepts for grades 3-5. In this case, I didn’t change as much of the original problem because the context is so simple. Here are the 3 slides I created to scaffold presenting the information:

Slide 1

Slide 2

Slide 3

It’s important to remember that the power of a numberless word problem lies in the conversation students have as you reveal each new piece of information. That conversation is driven by the questions you ask as more and more information is revealed. Here are sample questions you could use as you discuss each slide of a numberless word problem:

• What do you know?
• What information have you been given?
• What do you understand about the information given?
• What kind of problem could this be?
• What information do you know now?
• Does this new information help you?
• What does the new information tell you?
• How does the new information change or support your thinking?
• What operation(s) does this situation make you think about?
• What kinds of questions could be asked about this situation? (This can be asked on several slides, not just the final one.)

The fun part for this particular situation was thinking of all the different questions I could ask:

So there you have it – three very different examples of numberless word problems. As cool as I think numberless word problems are, please note that not every problem needs to be a numberless word problem. We have to be intentional about when and how much we provide scaffolding to our students. However, knowing about this type of problem is a great tool to have in your belt when you’re looking for ways to help your students develop a deeper understanding of the mathematical relationships in real life situations.

If you have any questions, please don’t hesitate to ask in the comments!

[UPDATE – I’ve made a page on my blog devoted to numberless word problems. Check it out for more resources.]