Category Archives: Math Teaching

The Annotated Numberless Word Problem

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:

  1. 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.
  2. 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.

Image Source: https://pixabay.com/en/ice-ice-cream-milk-ice-cream-waffle-2367072/

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:

  1. 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).
  2. It serves as a pay off for all of the hard work students just did to make sense of and solve the problem.
Here’s a link to the video I included in this 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.

Final thoughts

Now that you’ve seen me put together this numberless word problem in pieces, here’s your chance to see the finished product. This link will take you to the slideshow for the finished product.

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.

You Didn’t Hear It From Me

On Thursday, I’ll be sharing about numberless word problems at the NCTM Annual Conference in Washington, D.C.

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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.”

Kristine Venneman, Elementary Mathematics Specialist, Middletown:

“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!

A Little Preview

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.

2018-NCTM-Program-Bushart

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!).

The Collie and Chihuahua Problem – This is a comparison problem where the difference is unknown.

The Ancient Penguin Problem – This is another comparison problem. This time the larger quantity is unknown.

The Sand Castle Problem – This is an equal groups problem with an unknown product.

The Minecraft Problem – This is a multi-step problem involving multiplication and addition.

The Piano Practice Problem – This is a multi-step problem involving addition and subtraction.

The Pie Problem – This is a multi-step problem involving multiplication.

Enjoy! And if you’ll be joining me next week at NCTM, I look forward to seeing you in Washington, D.C.!

Inspiration – Summer Edition

As you may or may not know, I have a tendency to roam the seasonal aisle at Target, looking for mathematical inspiration. So far I’ve shared photos I’ve taken at Halloween, Valentine’s Day, and Easter. You can find them all here.

Today I was stopping by Target for some bug spray which just so happens to be next to the summer seasonal aisle. I couldn’t resist the urge to take a stroll and take some pictures. Here’s what I’ve got for you today.

How many large wooden dice are in the package?

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It’s totally obvious, right? For younger students, maybe not so much. But even after everyone is in agreement that it’s 6, what do you think they’re going to say once you reveal the answer?

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Not what you were expecting, is it? You probably thought I was wasting your time starting with such a simple image. So now you get to wonder, “Why/How are there only 5 dice in this package?” Perhaps this will help:

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That burlap bag has to fit somewhere!

Let’s move on to another large wooden product. How many dominoes are in this pack?

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It might be a little hard to tell from this perspective. Let’s look at it another way.

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Barring any more burlap sacks, you might just have the answer. Before we find out, stop and think, what answers are reasonable? What answers are not reasonable?

Ok, time to check if you’re right.

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No surprises here. Although after the first image, I probably had you second guessing yourself. There’s something to be said about the importance of how we sequence tasks.

Speaking of sequencing tasks, let’s move on to another one. How many light bulbs on this string of lights?

20170601_191802

I really like this box because you get this tiny 2 by 3 window, and yet it’s such a perfect amount to be able to figure out the rest. This would be one I’d love to give students a copy of the picture and let them try to show their thinking by pointing or drawing circles on it.

Again, this is a great time to ask, what answers are reasonable? What answers are not reasonable? Assuming the light bulbs do create a rectangular array, there are definitely some answers that are more reasonable than others.

After some fun discussion about arrays, it’s time to check the actual amount.

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So fun! Like I said, I love this image. Let’s look at another package that caught my eye.

How many pieces of sidewalk chalk in this box?

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I was pleasantly surprised to find that Crayola put arrays on top of all their summer art supplies. It’s like they were designed to inspire mathematical conversation! Granted, the box doesn’t give it away that the dots represent the pieces of chalk, I wouldn’t point it out to students. I’d let them wonder and make assumptions about it. It’ll turn out that their assumptions are completely right, and how satisfying that will be for them!

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Since we’re talking about arrays, which means we’re talking about multiplication, let’s shift gears a bit to look at some equal groups.

How many plastic chairs in this stack?

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And to throw a wrench into what looks to be a simple counting exercise, how much would it cost to buy the whole stack?

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Now students have got some interesting choices about how they calculate the cost. The fact that half the stack is blue and half the stack is red is just icing on the mathematical discussion cake.

My final image from the summer seasonal aisle has been a real head scratcher for me.

How many water balloons do you estimate are in this package?

WaterBalloon

 

What is an estimate that is too low?

What is an estimate that is too high?

What is your estimate? How did you come up with that?

Take a look at the box from another angle, and see if you want to revise your estimate at all.

WaterBalloon2

 

We clearly have groups – eight of them to be precise – but the question I’m not entirely sure about is whether there are eight equal groups. Maybe? And if there are equal groups, then there are certain answers that are more reasonable than others.

So how do you wrap your head around this?

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I’ll give you a moment to think about why this is confusing me a bit.

Assuming there is an equal amount of each color, this doesn’t make any sense! But then I noticed the small white tag on the set of purple balloons.

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Oh! That explains it. There’re only 260 balloons in here so…no, that still doesn’t work if these are eight equal groups.

Oh, then maybe it’s 5 more than 265 so it’s actually 270 so…no, that doesn’t work either. So I’m left to conclude that either this is not a pack with eight equal groups or there is some funny math going on! Sadly, $25 is a bit steep to satisfy my curiosity. If any of you purchase this pack and want to count balloons, I’d love to get the full story.

And with that, my tour of the summer seasonal aisle comes to an end. If you’re just finishing the school year, bookmark this post to revisit when school gets back in session. What a fun way to start the year! If you’re still going strong, then I hope you’re able to use these to spark some fun, mathematical discussions in your classrooms.

 

 

The Slow Reveal

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.

Kassia01

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.

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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.

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Once she included some numbers, my first instinct was that this graph is about ages. Apparently I wasn’t alone in that assumption!

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And then there’s the final reveal.

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So why did Jenna create and share this graph? What was her mathematical goal?

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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.

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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!

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.

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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?

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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?

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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?

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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?

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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?

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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?

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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?

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What is this question asking?

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

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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!

Decisions, Decisions

This week our Math Rocks cohort met for the fourth time. We had two full days together in July, and we had our first after school session two weeks ago. One of our aims this year is to create a community of practice around an instructional routine, specifically the number talks routine. We spent a full day building a shared understanding of number talks back in July. You can read about that here. We also debriefed a bit about them during our session two weeks ago.

This week we put the spotlight on number talks again. We actually broke the group up by grade levels to focus our conversations. Regina led our K-2 teachers while I led our 3-5 teachers. The purpose of today’s session was to think about the decisions we have to make as teachers as we record students’ strategies. How do you accurately capture what a student is saying while at the same time creating a representation that everyone else in the class can analyze and potentially learn from?

We started the session with a little noticing and wondering about various representations of 65 – 32:

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Very quickly someone brought up exactly what I was hoping for which is that some of the representations show similar strategies but in different ways. For example, the number line in the top left corner shows a strategy of counting back and so do the equations closer to the bottom right corner.

This discussion also led into another discussion about the constant difference strategy – what it is and how it works. It wasn’t exactly in my plans to go into detail about it this afternoon, but since my secondary goal for the day was to focus specifically on recording subtraction strategies, it seemed a worthwhile time investment.

After our discussions I shared the following two slides that I recreated from an amazing session I attended by Pam Harris back in May. (For the record, every session I attend with her is amazing.)

The first slide differentiates strategies from models. Basically, if you have students telling you their strategy is, “I did a number line,” and you’re cool with that, then you should read this slide closely:

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The second slide differentiates tools for building relationships from tools for computation. This slide is crucial because it shows that while we want students to use tools like a hundred chart to learn about navigating numbers within 100, the goal is to eventually draw out worthwhile strategies, such as jumping forward and/or backward by 10s and then 1s.

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The strategy on the right that shows 32 + 30 followed by 62 + 3 is totally the type of strategy students should eventually do symbolically after building relationships with a tool like the hundred chart.

After blowing their minds with those two slides, I led them in a number talk of 52 – 37. During my recording of their strategies, I stopped a lot to talk about why I chose to do what I did, to solicit their feedback, and even to make some changes on the fly based on our discussion.

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For example, in the top right corner of the board I initially used equations to represent a compensation strategy. Someone asked if this could be modeled on a number line because she thought it might make more sense, so I did just that in the top left corner. By the time we were done they were like, “Oh, hey! That ends up looking like a strip diagram!”

It was amusing that the first strategies they shared involved constant difference. They were so excited about learning how the strategy worked that they wanted to give it a try. I didn’t want to quash their excitement by telling them that the strategy tends to work better, especially for students, when you adjust the second number to a multiple of ten. I wanted to stay focused on my goals for the day. We’ll discuss the strategy more in a future session.

(Unless you’re in Math Rocks and you’re reading this! In which case, see if you can figure out why that’s the case and share it at our next meeting.)

After some great discussion about recording a variety of strategies, we watched Kristin Gray in action leading a number talk of 61 – 27.

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We talked about how she recorded the students’ strategies. We also talked about some really lovely teacher moves that I made sure to draw attention to.

We wrapped up our time together talking about what new ideas they learned that they wanted to try out with their students. I had asked one of the teachers to lead us in another number talk, but we ran out of time so I think I’m going to have her do that at the start of our next session together. Hopefully everyone will have had some intentional experiences with recording strategies between now and then to draw on during that number talk.

Oh, another thing we talked about at various points during the session was how to lead students in the direction of certain strategies. This gets into problem strings, which may or may not happen in number talks depending on whom you talk to. Regardless, here are some we came up with. Can you figure out what strategies they might be leading students to notice and think about?

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