035: Sense-Making with Numberless Word Problems

It might sound strange at first. After all, how can you solve a math problem without any numbers? But that’s exactly what makes numberless word problems such a powerful tool. In this episode, we explore how removing numbers actually helps students build deeper comprehension, plan solution paths, and develop the habit of asking, “Does this make sense?”

We’ll unpack why students often rush into calculations without understanding the context, and how numberless word problems shift their focus back where it belongs: on sense-making. Plus, I’ll share simple ways you can start using this strategy with the materials you already have.

Hello, Meaning-Makers! Jillian here from Team Meaningful Math. Welcome to this episode of the Meaningful Math Podcast. I’m excited to explore one of my favorite problem-solving tasks: numberless word problems! Now, I know what you might be thinking… how could a MATH problem be NUMBERLESS? Don’t worry – keep listening, and I’ll explain. Let’s get started! 

Student Answers to an Unanswerable Problem

Robert Kaplinksy, one of my favorite math gurus, has a video on YouTube titled “How Old is the Shepherd?”. In the video, Robert posed a question to 32 eighth-grade students individually and observed how they responded. Here’s the question he asked: 

“There are 125 sheep and 5 dogs in a flock. How old is the shepherd?” 

Robert wanted to see how students responded to a question that wasn’t answerable given the information in the story. You would not believe the responses he received. 

Some students did 125 divided by 5 and said the shepherd was 25 years old. Some students did 125 – 5 and said the shepherd was 120 years old. A student even added 125 and 5 and said the shepherd was 130 years old. A few students didn’t do any math and simply stated an answer, such as, “50 years old” or “5 years old”. 

In fact, of the 32 students, 24 of them provided an answer for the shepherd’s age and thus were not able to make sense of the problem. Only 8 students declared that the problem didn’t make sense and was not solvable. 

A Compulsion to Calculate

Wow. Let’s think about what this means. Those 24 students were not able to make sense of the problem. Instead of truly thinking about the story, most of them launched into problem-solving mode right away by finding the numbers and doing something with them. They didn’t take the time to question the validity of the story. 

How often do our students do this? They read a story problem, pick out the numbers and then do an operation with those numbers. They may choose the operation based on the one they’re most comfortable with, key words they see in the story, or the operation that is the focus of their current math unit. They slap a label on their answer and call it done, without even stopping to think if the answer makes sense. 

This is often called “a compulsion to calculate”.  Instead, we want our students to be like the 8 students in Robert’s video who made sense of the story first before doing anything else. 

Making Sense of Problems

The ability for students to make sense of problems is so important that it is actually part of the first Math Practice: Make sense of problems and persevere in solving them. Sidenote – if you are unfamiliar with the Math Practices check out podcast episodes 1-8 for a deep dive into each practice.

In order for students to be able to make sense of problems, they must be able to do three things: 

1. Explain the meaning of the problem. 
2. Plan a solution path instead of simply jumping into a solution attempt. 
3. Continuously ask themselves, “Does this make sense?”

The students in Robert’s video who produced an answer for the shepherd’s age were unable to do these three things, especially the third one. Not only did the story not make sense, but those students’ solution paths did not make sense. 

There was nothing in the story that called for an operation to be used. Furthermore, many of the ages they produced did not make sense. Is it realistic for the shepherd to be 120, 130, or 5 years old? Definitely not. 

Using Numberless Word Problems

So, how can we help our students make sense of problems? One way is through the use of Numberless Word Problems. 

What is a Numberless Word Problem? Well, it’s pretty much what it sounds like – it’s a word problem with the numbers taken out. So instead of focusing on the numbers, the students are forced to focus on understanding the story. 

Steps of a Numberless Word Problem

Let’s talk through the steps of a Numberless Word Problem. 

1. Present a Numberless Word Problem to the class. This can be done by revealing one sentence at a time or revealing the whole problem at once. In addition to removing the numbers, you may even choose to remove the question. More on this later. 
2. Ask students to imagine the story, piece by piece or in its entirety, depending on if you reveal the whole story at once or show it sentence by sentence. One way to do this is to have students draw a picture of what is happening.
3. Facilitate a class discussion to help the students make sense of the story. Ask questions such as:
   1. What is going on in the story?
   2. What do you know for sure?
   3. What are you unsure about?
   4. What do we know about _______ (referring to quantities and relationships) even though we don’t see any numbers?
   5. What are you picturing in your mind?
   6. How does your picture show what’s happening in the story?
   7. How did your picture change? (if you choose to reveal one sentence at a time)
4. Add the numbers back into the story. If students drew a picture earlier, you can allow them time to adjust their pictures based on this new information. 
5. If you removed the question from the story, now is a great opportunity to ask students what they think the question could be. This is incredibly powerful. It’s truly the “icing on the cake” in terms of students making sense of the problem. 

I like to list out the possible questions students come up with and either choose one for them to answer or allow students to choose the question they want to answer. Students, especially those in the primary grades, might need support when it comes to thinking of what the question could be. 

If students are continuously asking off-topic questions, ask them to come up with a “math question”. You might also find that your students say something like, “The question could be ‘What is 3 + 5?’” This is normal! It will take time and guidance to help students formulate their ideas into questions, but this work is worth it. 

1. Allow students time to solve the problem and then share their work with others. Make sure to allow time for students to justify their thinking, defend their answers, and critique the reasoning of others. 

Investment of Time Well Spent

Now, this might all sound great, but you may be worried if this will take too much time. Yes, it will take more time than a standard word problem. Keep in mind, not every word problem you do has to be numberless. But, when you choose to do a Numberless Word Problem, know that you will probably get through one problem in place of several. But it is time WELL SPENT because your students will be deeply involved in making sense of the problem. 

This work will pay off in spades when students encounter word problems that aren’t numberless because they will have experience and knowledge of how to make sense of problems. I’ve even seen students cross out numbers when given a standard word problem after they’ve had a lot of experience working with Numberless Word Problems! They work to make sense first, and then go back to the numbers after.

If you’d like to get started with using Numberless Word Problems in your classroom, I encourage you to begin by taking existing word problems that you have in your curriculum and simply removing the numbers. It’s that easy! There’s no need to reinvent the wheel. You can take the word problems that you already have, remove the numbers, and voila – you have numberless word problems! 

Friends, I hope this episode gave you some insight as to the importance of helping our students make sense of problems and how Numberless Word Problems can help us do that. 

If you enjoyed this episode, please take a few seconds to leave us a review! We’d love to hear from you, and your review will help bring this information to even more teachers and impact even more students. Thank you for tuning in! Until next time, Meaning-Makers, Have a great one!