# 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.

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!

# Talking Up Talking Points

I have been talking up Talking Points ever since I got home from Twitter Math Camp in July. Don’t know what Talking Points are? No problem. You can learn more about them on the Twitter Math Camp wiki. We had a group led by @cheesemonkeysf who dove deeply into this topic and shared their work with the rest of us. When you have a chance, I suggest reading the document titled About Talking Points. In the meantime, here’s a brief summary:

Talking Points are simply a list of thoughts. They are statements which can be factually accurate, contentious, or downright wrong. They can be thought-provoking, interesting, irritating, amusing, smart, simple, brief or wordy.

That definition aside, it’s not so important to know what Talking Points are as it is to know what to do with them and why.

Often children (and even adults) are asked to discuss their ideas or work together in groups, but it quickly becomes apparent that those involved don’t really know what is being asked of them.

You have those participants who love to talk, but they don’t really know how to consider other people’s views. On the other hand, you have participants who are quiet. They find it hard to join in the conversation.

The role of the teacher is to make explicit the kind of talk that is useful, and that is where the Talking Points activity comes in.

The activity stimulates speaking, listening, thinking, and learning. It offers ways in to thinking more deeply about the subject under discussion. It gives everyone a chance to say what is on their mind, so that others can decide whether they agree or disagree.

If you want to try out the activity, you need to create small groups of about 4 people per group. Give each group a list of Talking Points. (You can write your own, or you can start by using some of the ones on the Twitter Math Camp wiki.) The group will engage in 3 rounds per Talking Point.

Round 1

• One person reads a Talking Point
• Go around the group. Each person says whether they AGREE, DISAGREE, or are UNSURE about the statement AND WHY.
• The most important part of this is that there is NO COMMENT by anyone else in the group. Their job is to listen.
• After everyone has had their turn, proceed to round 2.

Round 2

• Go around the group again. Each person says whether they AGREE, DISAGREE, or are UNSURE about their own original statement OR about someone else’s statement they just heard AND SAY WHY.
• Again, there should be NO COMMENT from anyone else in the group while someone is speaking.
• After everyone has had their turn, proceed to round 3.

Round 3

• Go around the group one final time. Each person simply states whether they AGREE, DISAGREE, or are UNSURE about the original statement. The group takes a tally and moves on to the next Talking Point.

When time is up, the facilitator can choose whether to have the participants complete a group self-assessment. (Check out the wiki for an example.) This gives them a chance to reflect on the discussion and how their group worked together.

The facilitator can also do a whole group debrief. I highly recommend doing this because it helps the participants reflect together, and it also gives the facilitator a chance to point out and emphasize to everyone behaviors and ways of talking that were effective. Here are sample questions that could be discussed:

• Who in your group changed their mind about a Talking Point? How did that occur?
• Who in your group encouraged someone else? How did that benefit the conversation?
• Who in your group provided an interesting additional idea and what was it?

You might be thinking to yourself, “Oh this activity is just like ___.” I’ve had several people tell me that, but once they have participated in their first Talking Points activity, they see how much deeper the conversation is than in the other activities they were thinking of.

The first time I tried this activity was with our district interventionists. This school year we are leading PD sessions with them about 10 times spread across the year, and one of the recurring themes is knowing their learners in general and using strategies to foster a growth mindset in particular. Before we dove into this topic, we created some Talking Points to get them talking with one another about their beliefs about intelligence and learning. Here is the set we used:

Talking Points About Intelligence and Personal Qualities

1. Your intelligence is something very basic about you that you can’t change very much.
2. You can learn new things, but you can’t really change how intelligent you are.
3. No matter how much intelligence you have, you can always change it quite a bit.
4. You can substantially change how intelligent you are.
5. You are a certain kind of person, and there is not much that can be done to really change it.
6. No matter what kind of person you are, you can always change it substantially.
7. You can do things differently, but the important parts of who you are can’t really be changed.
8. You can always change basic things about the kind of person you are.

The Talking Points were successful in getting the interventionists to start talking about ideas of growth mindset without us have to tell them anything. Many of them already felt strongly that intelligence can change (which made me happy considering the population they serve!) but I appreciated the discussion in a few of the groups where one person was able to point out some nuance in the language that made the others in the group consider the statement more carefully.

For example, in one group someone raised the point that she has an uncle with Down syndrome, and over the course of his life she doesn’t feel that his intelligence has really changed all that much. This personal experience made it difficult for her to fully agree with the Talking Points statements.

When the groups were finished discussing their Talking Points, we debriefed as a whole group. One of the recurring comments they made was how nice it was to have a chance to give their opinion and feel like it was heard by the rest of the group. Because of the “No Comment” rule, they knew that no one was going to interrupt them, and if someone tried, the other group members quickly reminded them of the rule.

A few weeks later, I was asked to share Talking Points with our district leaders at a Vertical Leadership Team meeting. This meeting consisted of all of the principals from all of our elementary, middle, and high schools as well as many other district leaders, about 120 folks in total. No pressure! I was nervous about what they’d think, but happy to have the forum to share the activity.

As with the interventionists, I was using the activity to lead in to a discussion of growth mindset research, but this time I revised the statements because I wanted them to each feel unique. In the first set I used, I felt like several statements said the same thing but in different ways. Here are the statements I finally settled on:

Talking Points About Learning and Intelligence

1. You can learn new things but you can’t really change how intelligent you are.
2. You are a certain kind of person, and there is not much that can be done to really change it.
3. When you are learning something new, you should avoid making mistakes at all costs.
4. You are smart when you can complete tasks quickly and accurately.
5. The people who are the best in their field tend to be naturally good at what they do.

I only had 20 minutes to introduce Talking Points, share how to do them, have everyone in the room try them, and then debrief. It was a rush to get it all done, but it went smoothly enough. I definitely think 30-35 minutes is probably better for a first introduction.

I love walking around and listening in on different discussions while the groups are going through the Talking Points. While listening to the district leaders, I loved hearing one woman say, “Well, I did agree with the statement, but now I disagree because of what you two just told me. I just didn’t know that before.” It’s not required that anyone change anyone else’s mind during Talking Points, but it sure is powerful that it has that ability to happen based on just three short rounds of sharing opinions.

Someone in my department used Talking Points just over a week ago with a team of instructional coaches. While each grade level has some common ELA vocabulary that is used, the way it is used has not always been consistent across grades. Before the coaches started preparing for upcoming PD sessions, she had them go through some Talking Points about the vocabulary. She said it was such a quick and powerful way to gain clarity as a group before they started their planning.

Ever since I’ve shared Talking Points, it has started to slowly spread in my district, and I’m excited to see the creative ways it is used to deepen conversations among both students and staff.

One thing I’ve learned in my first month and a half at my new job is to be prepared for anything. A few weeks ago, my boss told me and the elementary ELA lead that we were going to be providing professional development to our district’s 107(!) interventionists. Not only that, but we would be providing them a total of 14(!) all-day PD sessions over the course of the school year.

On one hand, how cool is it that we get to work with every single one of our campus interventionists to create some shared vision around Response to Intervention and to help build their knowledge and skills of math and reading intervention?

On the other hand, holey moley! That is quite an undertaking on top of our other job responsibilities.

Just over a week ago, we held our first session. Thankfully over the course of planning for it (and getting feedback from interventionists who couldn’t believe they were going to be off campus 14 full days during the year) this adventure did come into better focus. For example, instead of having to plan 4 hour sessions each time, my partner and I only have to prepare 2-2.5 hour sessions on math intervention. In addition, instead of meeting 14 times during the school year, the higher ups knocked that number down to 9.

It’s still a big undertaking, no doubt about it, but it is feeling more and more manageable. On top of that, our first session went beautifully. Our primary goal, which I’m happy to report we achieved, was getting buy-in from the interventionists, many of whom had no idea this was even going to happen until a few days before the first meeting.  Now that the first session is over, I’m excited for the work ahead.

Since this is such a big project, I’m going to try to blog and reflect about it this year to see what I learn from it to apply to future endeavors. I also want to share my experiences in case anyone else out there can benefit from them.

The day started with all of the interventionists in a large group to hear from the higher ups about the purposes of these meetings and why they felt they are important enough to warrant so much time away from school. The district leads for RtI and dyslexia also went over some important changes that the interventionists needed to know about. When all of that was over, the interventionists broke up into groups to attend either a math session or ELA session.

Show time!

I opened the math session with everyone sitting in a community circle. My background is in a program called Tribes, and while I mostly used it with my students, the community building ideas apply to adults as well. The first thing I did was have everyone go around and introduce themselves, describe the types of intervention they provide on their campus, and then tell everyone a movie, book, or TV character that best represented how they felt at that moment.

I told them I felt like Lucy from The Lion, the Witch, and the Wardrobe, specifically the scene where she parts the coats in the wardrobe to reveal a newly discovered snowy landscape before her. I told them I felt like I knew what I was getting into with this job, but being asked to do these sessions this year opened me up to exciting new possibilities I hadn’t imagined. I added that I hoped nobody would be turned to stone or sacrificed on a stone table during the year.

As the interventionists went around the circle, I was impressed with how thoughtful, creative, and telling their responses were. Listening to them talk, I picked up on themes about wanting to be in control but feeling overwhelmed, about wanting to do the best job possible for their students, and about recognizing this opportunity to grow as a leader on their campus. While at first I had felt silly asking them to name a movie, book, or TV character, it ended up being a great way for them to share their feelings and realize that many people in the room felt the same way.

After introductions we moved into an activity called Talking Points. I learned about this activity in July from @cheesemonkeysf at Twitter Math Camp. I love Talking Points! They provide a way for people to improve exploratory talk, to dive deeper and have more meaningful conversation. You can download instructions and see examples of Talking Points on the Twitter Math Camp wiki.

The Talking Points that I had the interventionists do were all statements related to growth and fixed mindset. The statements didn’t use those exact words, but rather they got the groups talking about things like whether intelligence is something that can or cannot be changed. I did this on purpose because I wanted to know their current thinking on the matter. As interventionists, these people work with students who are having a difficult time in school. They not only need academic support, but they need someone who can help motivate them and encourage them to believe that they can learn. Like I told the interventionists, “If you didn’t believe that these kids can learn, then why would you bother showing up to work every day?”

Thankfully the interventionists seemed to believe in growth mindset by and large, so the groups tended to agree with each other, which I’m ultimately okay with. I was happy to see some dissenting opinions here and there though. Those groups were able to tease out some interesting ideas that the other groups missed out on.

After debriefing the Talking Points, I gave the interventionists a copy of our district’s math goals. I asked them to read the goals and do a quick write about how these goals are currently being met (or not) in their campus’ intervention program. Then they talked about their notes with the other folks at their table. This discussion was interesting because some of the interventionists didn’t even know we had district math goals. It also got some of them questioning whether their current intervention program was meeting the goals.

I used this discussion to segue into the foundation for our work this year, the What Works Clearinghouse guide on RtI: Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schools. The guide was put together by a panel including a research mathematician who is active in K-8, two professors of math education, several special educators, and a math coach. It provides 8 specific recommendations to help schools implement math intervention. The recommendations are based on the best available research evidence and the panel’s expertise in mathematics, special education, research, and practice.

Here are the 8 recommendations:

1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk.
2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8. These materials should be selected by committee.
3. Instruction during intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.
4. Interventions should include instruction on solving word problems that is based on common underlying structures.
5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas.
6. Interventionists at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.
7. Monitor the progress of students receiving supplemental instruction and other students who are at risk.
8. Include motivational strategies in tier 2 and tier 3 interventions.

All of the work we do this year is going to revolve around these 8 recommendations. To help kick it off, I had the interventionists look more closely at recommendations 2, 3, 4, and 5. I copied the sections about each of those four recommendations from the guide, and I had each interventionist read one of the sections. When they were done, they got together with the other people who read about the same recommendation. They collaborated to create two slides. The first slide contained up to 4 key points about their recommendation. The other slide contained noticings and wonderings their group had based on what they read. When the session was over, I put all of the slides together into one presentation that the interventionists could keep as a reference or that they could share with others on their campuses.

The interventionists were very receptive to what they read in the 4 recommendations we focused on in this session. They liked that the emphasis is supposed to be on number concepts rather than trying to keep up with what the teacher is doing in the classroom. Some of them said that by trying to fill gaps instead of focusing on key concepts they often feel like content mastery teachers rather than interventionists. They are hoping they can better define their role through our work this year.

We ended the session by revisiting our district math goals to see how they related to the 8 recommendations. It was very easy for the interventionists to see that these recommendations align very well with our district’s math goals. That’s not to say that their work with their students in meeting these goals will be any easier per se, but it is reassuring to know that the work they are doing with their students is going to be meaningful and supported by research.

All in all, the interventionists left excited about the adventure we’re embarking upon. They’re especially happy that they’ll have the opportunity to get to know each other better so they can utilize each other as resources. I couldn’t ask for a better outcome from our first time together. Part of me is even a little sad that I only get to plan 8 more sessions instead of 13.

Just a little. I still have plenty of other work to do.