Category Archives: Numberless Word Problems

Trick or Treat!

Now that I’ve completed sets of numberless word problems for all of the addition and subtraction CGI problem types, I wanted to do something fun.

This school year, my co-worker Regina Payne and I have been visiting the teachers in our Math Rocks cohort. One of the things they’ve been graciously letting us do is model how to facilitate a numberless word problems. In addition to making this a learning experience for the teachers, we’ve made it a learning experience for ourselves by putting a twist on the numberless word problem format.

Instead of your usual wordy word problem, we’ve been trying out problems that include visuals, specifically graphs. Instead of revealing numbers one at a time, we’ve been revealing parts of the graph. Let me walk you through an example I made tonight.

Here’s the graph I started with. I created it with some data I found on the Internet.

graph06

If I threw this graph at a 4th or 5th grader along with a word problem, they would probably ignore what the graph is all about and just focus on getting the numbers they need for doing whatever computations they’ve decided to do. They would probably also ignore a vital piece of information – the scale that says “In Millions” – which means their answer is going to be about 1,000,000 times too small.

But what if we could change that by starting with something a little more accessible like this?

graph01

What do you notice? What do you wonder?

I’m guessing at least one student in the class would comment that it looks like a bar graph. Interesting. What do you think this bar graph could represent?

Oh, and you think a bar is missing in the middle. Interesting. What makes you say that?

graph02

What new information was added to the graph? How does it change your thinking?

Oh, so there is a bar between Hershey’s and M&M’s. How tall do you think the bar for Snickers might be? Why do you say that?

graph03

Now we know how tall the bar for Snickers is. How does that compare to our predictions?

Considering everything we know so far, what do you think this bar graph is about? What other information do we need in order to get the full story of this graph?

graph04

What new information was added to the graph? How does it change your thinking about what this graph is about?

What are Sales? How do they relate to candy?

What does “In Millions” mean? How does that relate to Sales?

I know we don’t have any numbers yet, but what relationships do you see in the graph? What comparisons can you make?

graph05

What new information was added? How does it change your thinking?

Hmm, how many dollars in sales do you think each bar represents? How did you decide?

graph06

How do the actual numbers compare to your estimates?

What were the total sales for Reese’s in 2013? (I’d sneak in this question if I felt like the students needed a reminder about the scale being in millions.)

What are some other questions you could use answer using the data in this bar graph?

graph07

What is this question asking?

How can you use the information in the graph to help you answer this question?

*****

I may or may not actually show that last slide. After reading this blog post by one of our instructional coaches Leilani Losli, I like the idea of letting the students generate and answer their own questions. In addition to being motivating for the students, it makes my time creating the graph well spent. I don’t want to spend a lot of time digging up data, making a graph, and then asking my students a whopping one question about it! That doesn’t motivate me to make more graphs. I  also want students to recognize that we can ask lots of different questions to make sense of data to better understand the story its telling.

Some thoughts before I close. This takes longer than your typical numberless word problem. There are a lot more reveals. Don’t be surprised if this takes you at least 15-20 minutes when you take into account all of the discussion. When I first do a graphing problem like this with a class, it’s worth the time. I like the extra scaffolding. Kids without a lot of sense making practice tend to be pretty terrible about paying attention to details in graphs, especially if their focus is on solving an accompanying word problem.

If I were to use this type of problem more frequently with a group of students, I could probably start to get away with fewer and fewer reveals. Remember, the numberless word problem routine is a scaffold not a crutch. My hope is that over time the students will develop good habits for attending to features and data in graphs on their own. If you’re looking for a transition to scaffold away from numberless and toward independence, you might start by showing the full graph and then have students notice and wonder about it before revealing the accompanying word problem.

If you’d like to try out this problem, here’s a link to a slideshow with all of the graph reveals. You’ll notice blank slides interspersed throughout. I’ve found that if you have a clicker or mouse that has a tendency to jump ahead a slide or two, the blank slide can prevent accidental reveals. It also helps with graphs because when I snip the pictures in they aren’t always exactly the same size. If the blank slides weren’t there, you’d probably notice the slight differences immediately, but clearing the screen between reveals mitigates that problem.

Happy Halloween!

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.

CGI

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!