I was recently invited on the Complementary Angles podcast to share about numberless word problems. Over the course of an hour we talked about:
The research that supports the use of numberless word problems
How numberless word problems empower all learners in working with problem situations
Advice for teachers implementing numberless word problems for the first time as well as advice for veteran users
How numberless word problems have evolved over time
How numberless word problems can be used in remote learning environments
That’s a lot of ground to cover, so they broke up the interview into two episodes. Check them out below.
Part 1:
Part 2:
If you have any advice for teachers using numberless word problems, particularly if you’ve been using them during remote learning, share them in the comments and/or tweet them using the hashtag #numberlesswp. We’re all better together!
I recently modeled a numberless word problem in a 4th grade classroom. A few weeks later, I got an email about how the teachers were attempting to create and use some of their own, but they were encountering a problem…writing their own problems was harder than they thought!
They reached out to me for support, and I thought I’d share with you what I shared with them in case it’s helpful to anyone else creating their own numberless word problems.
1. Start with a problem
First things first, start with the problem you want to transform into a numberless word problem. Here’s the problem I started with for this example:
I type the problem on a slide, either in Powerpoint or Google Slides. You can create your problem on chart paper or on strips of paper if you’re working with a small group. I’m partial to digital slides because of some other features you’ll see later in the post.
2. Work backward
From here I create a copy of this slide and remove some of the information. Usually I start by removing the question.
Next I copy this new slide and again decide what information to remove. In this case I decided to remove the entire last sentence. That sentence dramatically changes our understanding of the situation. If you look at the slide below you’ll see that we know the total number of kids eating ice cream and the number of kids eating chocolate ice cream.
The situation is very open right now. The rest of the kids could be eating a variety of different flavors – vanilla, strawberry, chocolate chip. When I reveal the sentence that the rest of the students are eating vanilla ice cream, there’s a nice element of surprise because you aren’t necessarily expecting that the kids are only eating just two different flavors.
My next step is to remove one of the numbers. In this case I’ll take away the number of children eating chocolate ice cream.
Finally, I’ll remove the number in the first sentence to get me to the beginning of this problem. This is the first text students will read.
I structure my slides to minimize changes. I don’t want to overwhelm the students by revealing too much all at once. I will add new sentence, but I avoid changing language that’s already on the slide, if possible. More often than not I’m only changing a word like “some” into a specific quantity. There are rare instances where I’ll have to adjust a sentence as new information is added, but I try not to do that. I want the sentence structure to stay the same so that when the numbers are added that’s the only real change.
You might have noticed that I don’t include pictures on the slides with the text. This is intentional. I used to include pictures, but a colleague shared how distracting the pictures were for her students. Students were looking for meaning in them when they were only there essentially as decoration, with the intent that they would support visualizing. However, the pictures ended up confusing her students rather than helping because the students kept trying to make connections between the pictures and text. Since then I’ve avoided pictures on the text slides unless the picture is absolutely necessary.
3. Plan purposeful questions
The first step was to work backward to plan out each slide so that information is slowly revealed on each slide. Now it’s time to plan the questions I’m going to ask the students at each step along the way. I have two primary goals that I strive for in my questioning:
I want students to visualize what the story is about as it unfolds. If they’re not “seeing” it, then they’re likely not making much sense of it.
I want students to make guesses and estimates about quantities in the story using what they know about the situation and the relationships provided. I want them reasoning all along the way so that by the time they get to answering the question they are holding themselves accountable if their answer doesn’t make sense.
So now I go back through the slides in the order they will be presented and add the questions I plan to ask along the way.
Slide 1
Ask for a volunteer to read the story.
What are you picturing in your mind? What do we know so far? How many kids could be eating ice cream? How many kids could be eating chocolate ice cream? Why do you say that?
Have students draw a quick sketch of the story so far.
Slide 2
Ask for a volunteer to read the slide.
What do we know now that we didn’t know before? What does this tell us about the number of kids eating chocolate ice cream?
When a new slide is presented, I always ask a question to get students to state the new information. I’ve also worded this as, “What changed? What do we know now that we didn’t know before?”
Slide 3
Ask for a volunteer to read the slide.
What do we know now that we didn’t know before? How does this number compare to our guesses? Does it make sense? Are all of the kids eating chocolate ice cream? What could the other kids be doing?
Slide 4
Ask for a volunteer to read the slide.
What do we know now that we didn’t know before? What does this tell us about the number of kids eating vanilla ice cream? How do you know?
Have students draw another quick sketch of the story so far.
What question(s) could we ask about this math story?
Slide 5
What is the question asking?
Do you have all the information you need to answer that question?
Let students work on solving the problem. Confer with students as they work to look for strategies you want to bring up with the whole class.
4. The beginning and the end
Something I’ve been doing for the past year with numberless word problems is bookending them with visuals to add a little more texture to the experience.
The beginning
The first thing I do is find a high quality image or two to show the students and have them chat about before we dive into reading any text. My go-to website for images is Pixabay.
I type in a word or phrase related to the story problem, like ice cream, and more often than not I hit the jackpot:
I look for a photo that I think will capture kids’ attention and activate their prior knowledge of the context. It allows students who may be less familiar with a situation to hear the relevant language, such as ice cream, chocolate, vanilla, and cone, before we dive into reading the text.
Here’s the picture I ultimately chose to engage students at the start of this problem, along with some notes of how I’d facilitate the opening discussion with the students.
What do you notice? What do you wonder? Give students 20-30 seconds of think/write time. Then let students share 1 noticing and 1 wondering with a partner. Finally let students share a few of their noticings and wonderings with the entire class. You may choose to record these in a t-chart, but it is not necessary for this problem.
Tell the students that today they are going to read a mathematical story about ice cream.
When I paste the picture on a slide, I always go into the Notes section of the slide and paste the source of the picture(s), usually the URL where I found it. On Pixabay, more often than not the photos have licenses allowing reuse. You can find the license information to the right of each photo. I know in the privacy of your own classroom it feels easy to get away with grabbing whatever picture you can find on Google Images, but it’s good habit to pull legal photos to avoid unforseen issues down the road. And with amazing sites like Pixabay and Wikimedia Commons available, there’s no reason not to at least start by looking for freely available photos.
The ending
I’ve been making it a habit to close each numberless word problem with a short video. This serves two goals:
It further builds students’ knowledge of the situation discussed. In the case of the problem I shared in this post, it was about kids eating ice cream so I found a short video of a kid making ice cream. Even if you can only find longer videos, you don’t have to show the whole thing. You could just watch the first minute (or whichever section is most relevant or interesting).
It serves as a pay off for all of the hard work students just did to make sense of and solve the problem.
I’m sure you can guess where I go to find videos. YouTube has such an endless supply of videos, that I haven’t yet encountered a situation where I couldn’t find a video worth sharing. Sometimes it’s the first video and sometimes it’s the tenth, but it’s always there waiting to be discovered.
In the Notes section on some of the slides, you’ll see references to students sketching in boxes. I created a recording sheet to try out when I modeled a different problem recently. If you want to check out the recording sheet, here’s the link. I don’t have a lot of experience using it yet so I don’t want to say more about it right now, but I do want to share in case it’s helpful.
If you have any questions, don’t hesitate to reach out in the comments or tweet me @bstockus. And if you create your own problem, please share it with us on Twitter using the #numberlesswp hashtag.
At this point it’s become an annual tradition that I make a batch of numberless word problems based on the results of the Safeway World Championship Pumpkin Weigh-Off. I’ve collected together the problems I’ve written based on the results from 2016, 2017, and now 2018 in this folder. [Update: Thanks to a request from my daughter, I’ve added some primary level pumpkin problems as well.]As with all the other files I share, you are welcome to edit them if you want to tweak them for your students. All you have to do is make a copy if you have a Google account or download the file in an editable format like Power Point. You will have full editing rights of your copy.
If you’re looking for some more mathematical inspiration as Halloween approaches, check out these three blog posts I wrote which include lots of photos and ideas for how to use them to spark mathematical conversations with your students.
On Thursday, I’ll be sharing about numberless word problems at the NCTM Annual Conference in Washington, D.C.
In preparation for my session, I reached out to fellow educators on Twitter, asking them, “How have numberless word problems impacted student sense making in your classroom?” I’ll have plenty to say about this in my talk, but I wanted to take this opportunity to let a wider variety of voices share their thoughts and reflections on using numberless word problems.
Macy, Math Interventionist, Arkansas:
“They no longer see two numbers and add. They think about the problem and what a reasonable answer could be.”
Julie Bourke, 2nd grade teacher, Michigan:
“I can actually watch my students shift from plucking out numbers and adding them to reading the problem and visualizing what is happening. This helps them solve and understand exactly what the quantities in the problem represent.
I also saw a shift in my repeated direction of “don’t forget the unit/label” has disappeared because the students aren’t thinking of the numbers as separate from the problem. They are making sense of the context, deciding on the best strategy to solve and the numbers in the problem aren’t really the focus.
This has also improved my own teaching. I was a “circle the number and underline the key words” teacher and I was teaching students to follow directions. Now I am teaching mathematicians who make sense of problems, develop strategies and discuss solutions within a context.
This has been an important shift for my career and my own understanding of teaching math.”
Kristen Mangus, Math Support Teacher, Maryland:
“I have shared these with teachers in my school, K-5. K teachers started using these when they began teaching word problem standards and they instantly noticed a difference in how their students solved problems compared to when they taught problem solving without numberless word problems.
Numberless word problems also reduce “number plucking” because students have time to think about the problem, make connections and ask questions so that they are ready and confident when the numbers are introduced.”
Kjersti Oliver, Middle School Instructional Facilitator, Virginia:
“These are great for MIDDLE SCHOOL TOO! Especially students that are EL or struggle with word problems! They can work for equation word problems, systems, proportions, etc.! Great entry point for students!”
Carrie DeNote, Math Interventionist, Florida:
“The student Notice/Wonder about everything now. I’ve seen N/W t-charts on their assessments where they have used it to help them make sense of a question.”
Jana Byrd, K-2 Elementary Specialist, Alabama:
“The very first time I used numberless story problems, I was amazed at the amount of math vocabulary that naturally surfaced during the discussion; greater than, less than, the same, equal, etc.
Without the numbers in the problem, I noticed that students focused on finding the relationship between the quantities even though they weren’t there. That prevented them from just grabbing numbers and doing something with them.
When the numbers were presented within the story problem, they did what made sense to them. They were able to decide on a strategy and discuss their thinking in a more clear manner. I’m sold on numberless word problems, especially when introducing new situations to students.”
Jordan Hill, 2nd grade teacher, Alabama:
“It has allowed the students to stop and make sense of the situation before attacking the problem.”
Wendy Wall, Mathematics Support Teacher, Virginia:
“Thank you! You have created an opportunity for students to talk and reason. You have created a resource teachers love!”
Deepa Bharath, Math Coach, Florida:
“Focus is on understanding the context, considering what is asked and possible strategies – students can notice structures and similarities, this is like the other one we did, when numbers are shown students tend to think less and just compute. Also helped students to be less afraid of fractions and large numbers – we solved the same problem with whole numbers before working with fractions, almost like a number string estimating first how the answer would be affected.”
Nicole Grygar, 1st grade teacher, Texas:
“When solving word problems, they are not jumping to conclusions. They are working all the way through the problem to make sure they are solving the right question.”
Christine Mauer, Special Education Resource & Inclusion Instructional Assistant, Texas:
“Taking the numbers out of the questions has allowed them to become immersed in the story first.”
Jenna Laib, K-8 Math Specialist, Massachusetts:
“Students are willing to think deeper and slower about world problems; they don’t shy away from a block of text as much, and they have a greater awareness of problem types (CGI style) which helps them determine their strategy. I have noticed the biggest change in students with disabilities, especially students with language-based disabilities like dyslexia.”
Melanie Tindall, Elementary Math Specialist K-5, New Jersey:
“Numberless word problems help students think about and visualize the problem. They help students think about what information they know and what information they need in order to solve the problem. They also help students think about what question(s) can be answered with the given information.”
“Students are essentially forced to consider the context to begin their solution path without simply adding or multiplying.”
Rose Scullion, K-5 Mathematics Specialist, New Jersey:
“Before numberless word problems became part of regular instruction students would take the numbers they saw in the problem, cross their fingers, have a hope and a prayer, and perform some type of procedure or algorithm, with no sense if they were correct or not. Now, students are relying more on visualizing the mathematical context, planning out their solutions, and choosing strategies to solve.”
Anonymous, Math Coach, Connecticut:
“The use of them have increased students focusing on the context and sense making.”
Shawna Velt, Special Education Math Consultant, Michigan:
“I share this strategy with special education teachers to support students in understanding word problems. We use cubes to model along with each step”
Brian Buckhalter, K-4 Math Coach, Mississippi:
“Traditionally, the “goal” of math class is to find the answer. Numberless word problems take the attention away from finding the (usually) one correct solution. Instead, they open the door for discussion among students to share their interpretations and reasoning about problems. Then the focus shifts from following steps or other procedures to reasoning, examining relationships, extending patterns, doing what “just makes sense” (as my students would say) and other hidden beauties of truly understanding mathematics.”
Thank you to everyone who took the time to share their feedback and experiences! It was so heartwarming to read how numberless word problems have impacted other classrooms across the country. As someone whose mission it is to help students develop identities as mathematical sense makers, it means a lot that this strategy has helped so many of you foster that with your own students.
And to those of you able to join me at the NCTM Annual Conference in D.C., I look forward to seeing you in a couple days!
Next week I have the privilege of presenting a session about numberless word problems at the 2018 NCTM Annual conference. Even if you don’t teach in grades 3-5, I still invite you to join us because there will be lots of ideas shared of interest to multiple grade levels.
During the session, I’ll be referencing a few numberless word problems used over the course of several months in a 3rd grade classroom in my district. I thought it might be fun to share them before my session so folks could take a peak (and possibly even try one or two of them out before my session!).
This past week my work life and my daughter’s school life came crashing together in the most wonderful way.
I.
On the way home from school on Thursday, she asked if we could practice “take away.” At first we practiced numerical problems like “What is 3 take away 1?” and “What is 5 take away 2?” Eventually I asked her if I could tell my problems in a story. The rest of the ride home we told “take away” stories. I told a few, and then she wanted it to be her turn:
“This one is sad. There were 2 cats and 1 of them died.”
“There were 6 oranges on the counter. A girl ate 2 of them and they died in her mouth.”
“There were 8 trees, and 3 of them got cut down.”
“There were 6 roads, and 2 of them fell down.” (I was able to figure out she was referring to overpasses because that’s what we were driving under at the time.)
Slightly morbid, but she’s 6 years old, so I roll with it, especially since she isn’t usually this chatty about anything related to school.
Anyway, as we were getting closer to home, I remembered that the math unit she’s currently in in school uses some numberless word problems, so I asked, “Have you ever had a problem about some geese and some of them stop to rest?”
(Stunned silence)
“How did you know that?!”
“What about a problem about a boy who checks out some books from the library and returns only some of them?”
(Stunned silence)
“Yes! How did you know that one!”
“Because I wrote them.”
“What do you mean?!”
“I’m the author of the take away stories you’ve been working on in math class.”
And thus our two worlds – my work and her school – came crashing together for the first time ever.
I’ve mentioned to her before that I work with and help teachers, but it’s always been in the abstract. Finding out that I was the author of specific problems she’s encountered in her classroom just blew her mind. She wanted to see some of them when she got home. Knowing she probably won’t always be this interested in my work, I was only too happy to oblige.
Daddy: How many feathers could he lose? Me: 7. Daddy: I think he lost 10. Me: He can't lose 10! Daddy: Why not? Me: He only has 8. You can't take away 10 feathers. pic.twitter.com/kg3hF0N8BU
Daddy: Want to see a video of another Kindergarten class solving the take away problem about the resting geese? Me: Yes! Was it at my school? Daddy: No, it was at another school. Me: Hey! We have that book in our classroom. And that one, too! pic.twitter.com/4zkPisXLso
As I was scrolling through the suggested unit plan to find the numberless word problems, I asked her about other tasks in the unit to see which ones she remembered. I asked about Bag-O-Chips, a 3 Act Task from Graham Fletcher, which was planned for the day after the numberless word problems, but she said she’d never seen it before. I have no idea how closely her teacher follows the unit plan, but lo and behold, the next day in the car when I asked what she did at school she said, “We did the bags of chips!”
Daddy: What did you do in math today? Me: We did the bag of chips one. There were supposed to be 4 yellow bags, but there were 3. We need 1 more.
We talked a little bit about the task in the car, and a little later as we finished up dinner I showed her the Act 1 video. Her eyes lit up. “That’s the video!”
We kept going back and forth between the image of what came in the bag and the image of what should have come in the bag. She happily used her fingers to figure out how many missing bags there were of each flavor.
I thoroughly enjoyed talking through the task with her, and what a pleasant surprise when she wanted to do another.
III.
I’m not one to pass up an opportunity talk about math with my daughter, so I quickly scanned Graham’s list of 3 Act Tasks to find one I know we didn’t include in our suggested unit plans. I settled on Peas in a Pod.
Source: https://gfletchy.com/peas-in-a-pod/
First, we watched the video and estimated how many peas would be in each of the pods.
“I think there are 3 in this one, 4 in this one, and 10 in this one. No, 13 in this one.” (She estimated from right to left in case you’re wondering.)
“Hmm,” I said, “I think 3 is a good guess for the first one. I think there might be 4 or 5 in the second one, and I’m going to agree with your first guess of 10 for the third one.”
Estimation is a new skill for Kindergarten students. I talk about guessing and she talks about being right. She thinks the goal is to be the person who guesses the correct (exact) amount. I’m going to keep talking about being close and reasonable because over time I know her understanding of what estimation is will develop and refine.
Then we watched the reveal video.
Source: https://gfletchy.com/peas-in-a-pod/
“I wasn’t right and you weren’t right!” She exclaimed.
“That’s okay. All of our guesses were pretty close, even though none of them matched the exact number of peas. I was surprised that this one only had 2 peas in it. I thought for sure there were more in there.”
“Me, too.”
“Hmm, I have another question for you. How many peas are there altogether?”
“Let me count.”
“I want to see if you can do it without counting on the picture. How many peas were in each pod?”
“8 and 7…and 2.”
“So how could you figure out the total?”
At first she tried using her fingers. She counted out 8 fingers, and then continued counting from there. I couldn’t really tell what she was doing, but at one point, after lots of ups and downs of fingers, she said, “18.”
Pretty close!
I didn’t say that though. Instead I said, “Hmm, I wonder if that’s the right amount. What other tool could we use to check your answer?”
She decided to get her Math Rack to check, and as a complete surprise to me she said, “Can you make a video of me?” Make a video of you solving a math problem? Why, of course!
Watching her first attempt, it was fascinating seeing her trying to keep track of two separate counts: (1) counting on from 8, “…9, 10, 11, 12, 13, 14,…” and (2) counting the 7 she was combining with the 8, “1, 2, 3, 4, 5,…”
It seems like she abandoned the double counting when she was so close to being done. I wonder if she sort of gave up and just continued counting to 18 since that’s what she had thought the answer was before.
I had a split second to think about how to respond. I didn’t want to confirm whether the answer was correct, and I wanted to see if she would be willing to try combining the three quantities again.
There was definitely a lot more accuracy when she separately modeled each quantity! I was impressed with the double counting she was attempting earlier, but in the end she was more successful when she could show each quantity separately and then count all.
It was a proud dad moment when she didn’t just accept 17 as the correct answer. She decided we should look at the picture of all the open pea pods to check. And, sure enough, when I held up the phone with the image of all the open pea pods, she was able to count all and verify that there were in fact 17 peas.
All in all, I’m over the moon. All year long I’ve asked her about school (and math), but up until now her answers have been fairly vague. (“I’m so surprised,” said no parent ever.) The most I’d gotten out of her before was that they did Counting Collections.
But now we’ve actually had a full blown conversation about the work she’s been doing in school, specifically activities I wrote or helped plan for our Kindergarten units. I’ve always loved talking about counting and shapes and patterns with my daughter since before she ever started school, but to have our worlds collide like this was really special. I enjoyed getting to share and talk about my work with a very different, and more personal, audience than I’m used to.
This year my colleague Regina Payne and I tried something new as we visited classrooms across our district – numberless graphs. Similar to a numberless word problem, you present a graph with no numbers and proceed to have a rich mathematical discussion as you slowly reveal more and more information on the graph. Early in the school year, I shared a Halloween-themed numberless graph, and I also wrote a blog post about it.
We briefly touched on this work in our session at the 2017 NCTM annual conference, and it’s been exciting to see my #MTBoS colleagues taking the idea and running with it in their schools! In case you don’t follow them – which will hopefully change after reading this post! – I want to share their work so you don’t miss out on all the great stuff they’re doing.
Kassia Wedekind
Kassia has written two wonderful blog posts about how she took our ideas and tinkered with them to create a data routine called Notice and Wonder Graphs. I like this name because it’s more inclusive than numberless graphs. When it comes to graphs, you might hide the numbers, but you could just as easily hide other parts of the graph first. It all depends on your goals and how you want the conversation to unfold. In Kassia’s first post, she shares this graph with students. Notice it has numbers, and little else.
Curious what it’s about? Then check out Kassia’s post. I’m betting you’ll be quite surprised when you reach the final reveal.
I will share this snippet from her post:
I love this routine for many of the reasons that I love Brian’s numberless word problems–it slows the thinking down and focuses on sense-making rather than answer-getting.
But I also love it because it brings out the storytelling aspect of data. So often in school (especially elementary school!) we analyze fake data. Or, perhaps worse, we create the same “What is your favorite ice cream flavor?” graph year after year after year for no apparent purpose.
I’ve decided to make it a goal to think more about data as storytelling, data as a way to investigate the world, and data as a tool for action. In my next two posts (YES, people! I’m firing the ole blog back up again!) I’m going to delve into the idea that we can use data to discuss social justice ideas and critical literacy at the elementary level. I’m just dipping my toe into this waters, but I’m really excited about it!
And Kassia did just that! So far she’s followed up with one post where her students noticed and wondered about a graph showing the percent of drivers pulled over by police in 2011, by race. I love how the graph sparked a curiosity that got her students to dive more deeply into the data. How often does a graph about favorite desserts or our birthday months spark much of any curiosity?
Jenna Laib
Jenna shared a numberless graph that immediately got me curious! This is one she created to use with 6th grade students.
I can’t help but notice a bunch of dots grouped up at the beginning with a just few outliers streeetttcchhiiing across almost to the very end.
Once she included some numbers, my first instinct was that this graph is about ages. Apparently I wasn’t alone in that assumption!
And then there’s the final reveal.
So why did Jenna create and share this graph? What was her mathematical goal?
I especially loved this observation about how her students treated the dot at 55 before they had the full context about what the graph is really about.
Chase Orton
Chase wrote a detailed post about how he worked with 2nd grade teachers to do a lesson study about interpreting graphs.
…there’s so many rich opportunities for meaningful student discourse about data. That is, if it’s done right. Most textbooks suck all the life out of the content. Students need to understand that data tells a story; it has contextual meaning that is both cohesive and incomplete. Students need to learn how to ask questions about data and to learn to identify information gaps. In other words, students need to learn to be active mathematical agents rather than passive mathematical consumers.
Chase walks you through the lesson he and the teachers created and tried out in three different classrooms. I love how he details all of the steps and even shares the slides they used in case you want to use them in your own classroom.
He closes the post with a great list of noticings and wonderings about continuing this work going forward. Here are a couple of them about numberless graphs specifically:
We need to give students more choice and voice about how they make meaning of problems and which problems they choose to solve. Numberless Data problems like these can be be a tool for that.
The missing information in the graph created more engagement.
A huge thank you to Kassia, Jenna, and Chase for trying out numberless graphs and sharing their experiences so we can all be inspired and learn from them. I can’t wait to see how this work continues to grow and develop next school year!
If you’re interested in reading more first-hand accounts of teachers using numberless word problems and graphs, be sure to check out the ever-growing blog post collection on my Numberless Word Problems page. I recently added a post by Kristen Acosta that I really like. I’m especially intrigued by a graphic organizer she created to help students record their thinking at various points during the numberless problem. Check it out!
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.
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?
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?
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?
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?
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?
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?
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?
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.
[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?
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.
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.
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.]
Teaching to the Beat of a Different Drummer 