[*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.

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!

gfletchyThis makes me very happy Brian! It will be a lot of work but very well worth it in the end. Thanks in advance my friend.

bstockusPost authorMy pleasure! It’s very motivating to create and share when I know there are a lot of people who might benefit. Here’s to the work ahead! 🙂

howardat58I came across a very old book, “Problems without Figures”, lost the link, and found this, based on it:

There are 300 problems in the original, and he has updated half the book.

bstockusPost authorThank you for sharing! I think you shared this link for the original before:

I look forward to checking out both.

Deborah NicholsThank you! This is just what I needed to move forward!

bstockusPost authorSo happy to hear that! 🙂

Jamie DuncanI can help you out, Brian! I’m going to be needing compare problems pretty soon anyway. I will send them to you and you can decide whether or not to use them. I will look theough the CGI socumwnt first. Thank you for putting this together!

bstockusPost authorThat is awesome! Thank you so much, Jamie!

Sandy M.I was so excited to read this! We were just talking about how we needed to find a better bank of word problems. I had been using a set of word problems from Scholastic, called Think-Aloud Math Word Problems, which allowed for great discussion, but I love your plan to organize them by problem type! This will be so helpful! Thank you!

bstockusPost authorYay! I’m glad you like my plan to organize by problem type. I’m sure there are other ways I can organize problems in the future, but this feels like a good first step.

ChrisThis is fantastic and timely. RTI and math is a huge topic in my district (I’m facilitating a workshop this morning, in fact!) and I’ve been advocating using First Steps in Math, which is a wonderful resource, but somewhat cumbersome, therefore it doesn’t get used as much. Aside from RTI, you’ve provided a lot of information and ideas to share with teachers. Thanks!

bstockusPost authorMy pleasure! I shared this routine with our district interventionists last year, and they came back so excited about the conversations they were able to have with their students. At the time I hadn’t really thought about being so purposeful with them. Hopefully I’ll be able to share the ideas from this post with them and eventually the banks of problems so they’ll have them to support their work next school year.

Denise GaskinsSo cool! These will be fun to use with my elementary math club. Thanks 🙂

bstockusPost authorMy pleasure! I hope they enjoy talking about them and solving them!

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Jennifer BellHI Brian! I learned about your numberless word problems through Jaime Duncan. Wow! It really has helped my first graders in truly understanding what the context of the problem is and it has also helped me feel more confident in unpacking word problems along with the help of my students. THANK YOU for sharing your ideas, as well as the word problems (along with the notes) you are creating and sharing. I plan on sharing your information and blog with my teammates. My students have become much more successful in solving word problems using numberless word problems. I also agree with you about using the same type of word problem for 3-4 times before moving onto another type of problem. My students are talking and making connections and the room is full of nonstop productive math talk. Thank you for helping me on my journey in teaching math in a more purposeful and intentional way! 🙂

Jenn CarrHi Brian! Thanks so much for posting the numberless word problem banks! My colleagues and I learn so much from reading your blog. We have been trying out numberless word problems with multi-step word problems to help students make sense of the context and to support them in creating an equation to represent the problem. Any insight you can share on using numberless word problems with multi-step problems? Thanks!!!

Susan RyanThanks for sharing! This will be a valuable tool!

Ashley SkenterisI appreciate the emphasis on direct systematic instruction based on the structure of the problem. Using this tool will ease the creation of numberless word problems in my classroom.

Avery CameronThis is great! Thanks for sharing:)

Jeannette TerranovaI like the idea of building up the word problems progressively to increase the understanding as well as difficulty.

lindaI like how the questions progress towards a real application of math

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