Intentional Talk Book Study, Chapter 2

I don’t know that I would ever recommend reading Intentional Talk cover to cover. I know that’s exactly what I’m doing this summer, but I can see why it might backfire with regards to changing teacher practice. Across 6 chapters, roughly 112 pages, the book goes in depth into 6 different ways to plan for and structure classroom conversation. The material is so rich, I can see teachers feeling overwhelmed attempting to put all of it into practice in a meaningful way.

In fact, that happened this past year at an elementary school in my district. The faculty did a book study of Intentional Talk, but by the end of the year there was little evidence that teachers were leading varied and intentional discussions. They liked what they read, sure, but trying to plan for and implement all these discussion types got pushed to the side because of numerous other demands on their time. They suffered from the problem of biting off more than they could chew.

This is a defeatist way to start a post, but the reason I’m leading with these thoughts is because I’ve been thinking about how to share this book with teachers and begin to help them successfully incorporate these ideas into their practice. My answer is to start with chapter 2.

Chapter 1 is excellent, don’t get me wrong. It provides rationale for why well-planned discussions are important, but honestly that chapter is only a must read for someone who craves background information or a reason to pursue this work. If you’re reading the book, you’ve probably already decided you want to learn more about facilitating classroom conversations. You aren’t looking for convincing. In that case, you probably want to jump right in and learn something new. So my advice is to skip chapter 1 (for now, at least) and move straight to chapter 2.

And then stop.

Seriously. Quit reading the book.

Take the time to apply what you learn in this one chapter. Believe me, there’s plenty to sustain you for a while! Take time to establish your classroom norms. They are critically important to creating the safe, respectful environment your students need before you start tackling the other discussion types. Take time to teach students the talk moves. Students need practice in learning what to say and how to say it. Let them practice. Let yourself practice! The talk moves are probably going to be new to you, too. Practice using them until you feel comfortable with them. That will take the pressure off when you do finally decide to tackle one of the targeted discussion types.

How long should you wait before picking the book up again? I have no idea. But chances are you’ll know when you’re ready. It might take a couple weeks, a month, or it might take a full semester, but at some point you’re going to notice that your students have gotten really good at sharing and discussing their strategies for solving problems. You’re going to realize that you know the talk moves like the back of your hand. That will be the time to branch out and ask yourself the question, “What other kinds of conversations could we be having?”

At this point, I might recommend reading pages 1-5 in chapter 1 to reconnect with the book and the principles that guide it. When you’re done, consult the table at the bottom of page 3 that provides a brief summary of the goals of each discussion type. Read through those and think about which type might support your students where they are currently in their math learning. After you choose one of the targeted discussion types, read the corresponding chapter and then stop. Quit reading the book again, and take the time to practice the new discussion type while continuing to do open strategy sharing.

Assuming you’re like most teachers, you’re going to teach more than one school year. There’s no reason you have to learn and master all these discussion types in one school year. The most important thing you can do is start by creating a classroom culture where students’ ideas are valued by you and their peers. Where students feel safe taking risks in sharing their ideas because they know everyone in the class is there to support each other in making sense of mathematics. Chapter 2 will help you with this goal. The rest of the book is great, but it can wait. No rush.

Intentional Talk Book Study, Chapter 1

“Math discussions aren’t just about show-and-tell: stand up, sit down, clap, clap, clap.”

Designing and implementing quality mathematical discussions takes effort. It’s not as easy as just having students get up and share their answers to a problem. But don’t let that turn you away from working to improve your practice! This is just chapter 1, after all, and there’s still so much to read and learn. In this chapter the authors lay out four principles that should guide our classroom discussions:

  1. Each discussion should have a goal. This means thinking in advance what it is mathematically you want students to get out of the discussion.
  2. Be explicit! Students likely don’t come to you with the skills needed to participate in classroom discussions. The teacher’s role is to help the students learn what they should be sharing and how they should be sharing it.
  3. Students should be talking and responding to one another. It’s easy in a classroom “discussion” for all comments from students to be directed at you, the teacher. Instead the discussion should be a conversation amongst all the students around a particular mathematical idea.
  4. Students must believe that they can make sense of math, and their ideas are valuable, even when they aren’t fully correct. There’s a lot of learning to be found in mistakes, and we need to value those as much as correct answers. Getting students to share means they have to be willing to take risks, and as teachers our job is to make our students feel safe to do so.

In addition to presenting these guiding principles, this chapter also differentiates two types of classroom discussion: open strategy sharing and targeted discussion. Open strategy sharing is what many teachers already do to some degree in their classrooms. This is when you let students share their answers and solutions to a problem. The goal is to get a variety of responses out in the open. 

However, sometimes you have a particular mathematical goal you want to focus on, such as having the students justify why a particular strategy works. That’s when you would use a targeted discussion instead. These types of classroom discussion are much more nuanced and planning for each one is different. This is why open strategy sharing only has one chapter in the book while 5 chapters are focused on the different types of targeted discussion:

  • Compare and Connect – comparing similarities and differences among strategies
  • Why? Let’s Justify – justifying why a certain strategy works
  • What’s Best and Why? – determining the best (most efficient) strategy in a particular situation
  • Define and Clarify – defining and discussing how to use models, tools, notation, etc. appropriately
  • Troubleshoot and Revise – determining which strategy produces a correct solution or figuring out what went wrong with a particular strategy

The chapter includes three vignettes to help illustrate the differences between open strategy sharing and targeted discussion. I love how the authors insert comments about the intentional decision-making the teacher did throughout each conversation. It shows early on in the book that the teacher isn’t being herded into some lock-step approach. Rather, at every moment you have the power to guide and steer the conversation based on the needs of your students.

One thing that really stood out to me that I didn’t catch the first time I read this chapter is that it’s okay to stop a conversation and come back to it later. I know as a teacher I often let conversations run so long that I wouldn’t get to other things I had planned. In my mind the conversation was so great, it was okay that we were cutting into our next subject by 10-15 minutes. I think this speaks to how I wasn’t planning my discussions in advance. I just let them happen and let them run their course for as long as they were interesting. As the vignettes in this chapter show, however, important mathematical topics can be discussed over several class periods instead of trying to cram it all in to one sitting.

The other thing that stood out to me was how these discussions have the power to give a voice to all our students, not just the high achievers or the outspoken ones. Creating a sense of community where all ideas are valued and respected allows all children the opportunity to be heard and to demonstrate what they understand about math. As the authors say in the book, there are many different ways to be smart in mathematics:

  • making connections across ideas
  • representing problems
  • working with models
  • figuring out faulty solutions
  • finding patterns
  • making conjectures
  • persisting with challenging problems
  • working through errors
  • searching for efficient solutions

How much more exciting to look for and honor these skills in our students rather than seeking out just correct answers! Just think of what valuing these skills tells students about what it means to learn and do mathematics.

Intentional Talk: A Casual, Summer-Long Book Study

I started reading Intentional Talk way back in December. I loved what I was reading, but with many other demands on my time, I was only able to read a chapter here and there before leaving it by the wayside altogether.

Fast forward to now and what do I stumble upon? Some fabulous folks on Twitter are doing a book study throughout the summer, reading just one chapter per week. I can totally handle that! You probably can too! Here’s a flyer with all the details:

  
If you’re unfamiliar with the slow chat format, basically it means there is no set time to chat. Rather, during the week questions will be posted using the #intenttalk hashtag and you can read and respond whenever it fits your schedule. How convenient is that?

I will point out that the chat did start last week. So depending on when you get your copy of the book, you might have to play a bit of catch up, but don’t worry, you’re not that far behind. Here’s a schedule to show you how the reading is broken up week by week. Take note of the moderator next week for chapter 3, none other than Elham Kazemi, one of the authors of the book! How often do you get to take part in a book study where the author participates? This is awesome!

  
If you’ve gotten to this point and you’re thinking to yourself, I don’t even know what this book is about. Why should I even bother reading it? Good question!

In the book’s Foreward by Megan Franke, she lists numerous reasons why classroom conversations are crucial for mathematical learning:

  • Students achieve mathematically when they explain the details of their mathematical ideas, when they engage with the details of other’s mathematical ideas, and when others engage with their own mathematical ideas.
  • Engaging in mathematical conversations in productive ways can help students see themselves as smart and competent in mathematics.
  • Students learn to listen to others, ask insightful and respectful questions, and reflect on their own understandings.

Be prepared. This is challenging work, but it is also greatly rewarding work that is worth our time and effort. The authors lay out the vision, but they also provide support through vignettes from primary and intermediate classrooms, guiding principles to help you make decisions, and planning tools to help you get started. And with a community of folks participating on Twitter, you’ll have lots of support to ask questions and share ideas. I hope to see you throughout the summer!

November in June

Today marks the first day of CAMT 2015 (The Conference for the Advancement of Mathematics Teaching). If you’ve never heard of CAMT, then it means you probably don’t live in Texas. We’re such a large state, we have our own NCTM-like conference every summer.

The keynote I attended this morning – there were two going on simultaneously – was given by a guy named Alan November. I had never heard of him before today, but I’m happy I know who he is now.

His speech had two primary themes and both resonated with me. The first is that we should focus on moving beyond the classroom walls towards building networks. He gave the example of a first grade teacher named Ms. Cassidy. She not only has a blog, but she also has a Twitter account. Using social media, her class has connected with other classes from around the globe, and they share their learning with one another. 

One of the ways they do this is through  Math Talk Grade 1 (#mtgr1). Problems are posed to the hashtag, and the students tweet out their solutions through the teacher’s account. In some ways this isn’t special at all. Everyday in classrooms across the country teachers pose problems and students solve them. The difference? An audience! Students are able to see how students from all over the world solved the exact same problem! How cool do you think it is for a class of 7-year-olds to browse through answers from students in Canada? How much cooler is it for them to discover they have a different, possibly better, answer than a student in Italy?

Does this spark your curiosity? Want to try this out for yourself? You’re in luck! You can join the Global Math Task Twitter Exchange taking place during the 2015-16 school year. They’re looking for folks in grades K-12, so if you teach one of those grades, sign up on the Google doc linked on the site and enjoy! If you’re in grades K-5, tweet out to #ElemMathChat from time to time to raise awareness of this exciting opportunity.

The other theme from this morning’s keynote was the importance of students’ voices. It basically comes down to this, teachers, as experts in their field, know too much to fully understand the perspective of students learning the content for the first time. This is known as the curse of knowledge. But students? They can totally relate to one another since they’re all in about the same place educationally. And it just so happens that kids like to talk to one another.

How does this relate to education? One way is by having students create videos for each other explaining concepts, strategies, ideas, etc. Alan gave the example of a class where students were given the choice of creating a tutorial video for homework or the students could do a typical homework worksheet with 10 or so problems.

He shared the story of one student who chose to create a video which turned out to be only about 3 minutes long. When asked how long it took to make the video, the student said 3 hours! The student knew the regular homework assignment would have only taken 10-15 minutes to complete. Her reasoning for doing the video? Homework doesn’t help anybody. The teacher already knows the answers. She made the video instead because it would help her friends.

Alan also shared part of a keynote speech given by Shilpa Yarlagadda. She realized she only had access to her teachers during the school day. At night her resources were her boring textbook or videos made by adults who drone on and on for 10-15 minutes. Her solution? Create her own videos and share them on YouTube. What makes her work special is that in addition to sharing content, she was able to use stories to make it relatable to other kids.

It turns out there are lots and lots of student-created videos out there. One place to find them is Mathtrain.TV. Now, just because students are making their own videos, don’t think that the teacher can just sit back and do nothing. In fact, the role of the teacher becomes Editor-in-Chief. Unsurprisingly, kids make mistakes in their videos all the time, and without guidance their movies can frankly suck. Just because they can relate to their peers doesn’t mean they know how to produce engaging, accurate videos. They need the support of their teachers to point out mistakes and help them learn how to revise their work, similar to how teachers already support students when writing stories.

As a blogger and tweeter for nearly three years, I see the advantages social media and networking have had for me professionally, and I’m excited that more and more school districts are opening up these tools to students and teachers to use in their learning and work. They provide authentic audiences and leverage the ability of students to relate to one another on their own level.

Wrapping Up a Year of Math Intervention PD

Earlier this school year, I blogged about an exciting opportunity I had to offer PD to all 107 of our district’s interventionists spread across the entire school year. I anticipated blogging about the experience during the year, but life got in the way. Here we are at the end of the school year and I am about to offer my final PD of the year tomorrow morning.

At first I wasn’t sure what I would do with them tomorrow. The way the schedule worked, I haven’t had a session with them since February. And with it being the end of the school year, I wasn’t sure what theyd be interested in hearing about. More than anything, they’re probably counting down the days to summer like their students are!

I thought a lot about it yesterday and today, and I came up with a plan that I think does a great job of honoring the work we’ve done together this year, brings us full circle to where we began, and even hints at directions for next year. As with anything I plan, it won’t go perfectly tomorrow, so I want to capture my ideas tonight of how I’m imagining and hoping it will go.

I’m basically going to give the same session 3 times in a row for an hour and a half each. That’s the only way we can cycle through all of the interventionists and provide PD on math, ELA, and a third rotating topic. The trouble is that the first group in the morning is invariably late. The starting time is 8am, but I doubt I’ll have everyone until 8:15. The session post-lunch is just as bad. The only group that will maximize the hour and a half is my middle group from 10:10 to 11:40.

I have too much to cover, so I am planning to start with an activity that I can do with whomever is in the room when the session starts. I’m going to ask the folks in the room to brainstorm at their table all the different ways they can make 120 using base ten blocks. If you miss the activity, it’s okay because you might catch the sharing portion. Even if you miss that, you’ll actually see various representations in the reading we’ll move into after this activity.

All year long, the interventionists have been doing a book study of Kathy Richardson’s How Children Learn Number Concepts. It’s been a great book study because I didn’t require any reading outside of the sessions. Knowing how busy the interventionists are, I know many of them wouldn’t do the reading. Instead, I let them do all the reading during our sessions. Did it take a lot of time? Yes. However, it meant every single one of them got the chance to read and discuss the book. And this is a book you don’t want to miss!

Unfortunately because some of the planned sessions were axed from the calendar, we only got to fully read 4 of the 6 chapters. We don’t have time to read chapters 5 and 6, but I am having them read the introductions to each chapter which is 9 or so pages each. Chapter 5 is about understanding place value (hundreds, tens, and ones) as well as addition and subtraction. A key idea from this chapter is that students must understand how to decompose and compose numbers flexibly, which will serve the students when they add and subtract multi-digit numbers. The example from the chapter is composing 120 in a variety of ways using base ten blocks, just like in our opening activity.

 

  As people finish reading, I want them to solve 397 + 205 in two different ways. We’ll chat about the reading for a few minutes to reinforce the importance of unitizing, and then we’ll do a brief number talk of the addition problem. I specifically want them to connect their strategies to the concepts in the reading.

Then we’re going to jump over to watching some videos of students solving 1,000 – 998 from the Math Reasoning Inventory website. The first two videos show students who quickly recognize that the difference is 2. The third video shows Ana, a student who uses her finger to draw and solve the problem on the desk using the standard algorithm. After comparing and constrating the students, I want to show them two more videos of Ana solving 99 + 17 and estimating 18 x 21. In every video, Ana gets the question correct, but she relies completely on writing the standard algorithm with her finger. She does not evaluate the problems or demonstrate any number sense.

I specifically want to focus on Ana because I want to talk about how an emphasis on skill-building over sense-making with intervention students can result in students like her who can get correct answers, but they only have a strong grasp of procedures, not necessarily mathematics. I want to ask them how they think Ana is going to perform when she moves up into middle and high school math. Is there a point at which her procedural skills aren’t going to be enough?

After this we’ll move into the pre-activity for the next chapter in Kathy RIchardson’s book. I’m giving the teacher’s this problem along with three sample solutions.

“There are three fish in an aquarium. The middle-sized fish eats 2 times what the first fish eats, and the big fish eats 3 times what the first or little fish eats. If the first fish eats 3 pieces of food, how many pieces of food would the other two fish eat?”

One of the solutions correctly shows how much food the other two fish will get, while two of the solutions show misunderstandings about the multiplicative relationships in the problem. I’m not going to call anybody out here, but I do want the teachers to sort of self-check before they read whether they are able to recognize the multiplicative relationships in the problem.

This problem is talked about in chapter 6 of Kathy RIchardson’s book, which is about understanding multiplication and division. I liked this problem a lot because she shared how questions like this were given to students, and the results showed that 45% of the second graders in the study were able to think multiplicatively, but only 49% of the 5th graders were able to do so with ease. This implies that despite three years of learning about multiplication in grades 3-5, the students in 5th grade did not demonstrate any greater understanding of multiplicative thinking than students in 2nd grade who have had little to no formal study of multiplication or division.

I love this chapter because it demonstrates how all of the work students should have done learning how to unitize tens and hundreds lays the foundation for unitizing in multiplication and division. Students must be able to think of counting groups of things and thinking of each group as a unit. She gives the following example to demonstrate the subtle shift in thinking students need to make to move from additive to multiplicative thinking:

Rick: I figured out 8 + 8, and that was 16. Then I added 8 + 8 again.
Emily: I had two 8s and that was 16, and two more 8s make another 16. That means four 8s makes 32.

From here we’ll briefly talk about the results from a survey given to over 1,000 teachers who were asked the following two questions:

  1. What reading skills do you most often teach to skilled readers?
  2. What reading skills do you most often teach to less skilled readers?

The results are, sadly, not surprising to me, but having the data helps drive home the point that we have to be mindful of what we’re teaching to differnent populations of students. You can read more about this survey and see graphs of the results here.

At this point, I want to share our district math goals, which I shared during our first PD session back in September:

The K-12 RRISD mathematics standards articulate five general goals for all students:

  • That they learn to value mathematics,
  • That they become confident in their ability to do mathematics,
  • They they become mathematical problem solvers,
  • They they learn to communicate mathematically, and
  • That they learn to reason mathematically.

Just because a student is struggling in math and put in an intervention program does not mean that we should ignore these goals. If anything , we should strive that much harder to reach them with these students! Based on their struggles, these students are already at risk of not valuing mathematics and not being confident in their abilities. We should build up their skills, yes, but we should also help them find ways to value math, to learn how to use it for work and play.

To this end, I’m using a few imbalance puzzles I got from Sue VanHattums’ wonderful book Playing With Math, (they are originally from this blog post).

  
I’m giving the interventionists two to solve, though if they can figure out one, that might be all we have time for. I’m not advocating they give these to their kids, but I want them to experience something that requires some problem solving, but it’s also fun and gets you communicating with your peers. I want them to experience something that might inspire them to create some new interactions with their students that work toward our district goals.

I want them to feel like they are walking away with something they can actually use with their students, so I’m introducing them to open middle problems. I was already a fan of the website, but it was Michael Fenton’s recent Global Math Department session that put them back on my radar in a big way. If you’re unfamiliar with the open middle problem type, basically it’s a problem that has a lot of solution paths, though not necessarily multiple solutions. Here’s an example of one from the website:

  
This problem has one correct answer, but students can attempt to find it in a variety of ways. I’ll share a few more examples with them so they get a feel for why these can be powerful problem types to include at times in their work with their intervention students.

I’m going to wrap up the session by showing Megan Taylors Ignite Talk about replacing “teacher-proof” curricula with “curriculum-proof” teachers. I just saw this for the first time recently, and it resonated with me with regards to intervention. So often I hear the cry for “teacher-proof” curriculum materials for RtI that can be used with fidelity and “get results.” While I see why this appeals to some people, I don’t think it’s the right way to go. 

When it comes to response to intervention, the key idea to me is that the interventionists should be constantly responding to their students’ needs. This means having a skillset that allows you to adapt and customize as needed to help the children grow mathematically, not to follow some prescription as though we’re trying to cure a cold. My goal this year has been to help build that skillset in the interventionists so they feel empowered to do that with their students, and I want to end our yearlong PD journey on that note so they can venture into their summer feeling like they are confident and ready to take on the challenges of next school year.

The Fast and Furious First Grade

Despite being the elementary math curriculum coordinator for my district, I have rarely gotten the opportunity to visit classrooms this year. Today was one of those rare days, and I was so blown away by the two classrooms I observed. Hopefully I’ll have a chance to write about both of them, but tonight I’m going to focus on the first grade classroom.

All year long I’ve heard great things about this first grade teacher, so I was excited to finally have a chance to visit her classroom. When I came in, she had the students on the floor in front of her. She was sitting in a chair holding up seven fingers.

  

The students each had a small whiteboard on which they wrote all of the “facts” they could about the number the teacher was showing. After the students had a minute or two, she called one of the students up to the front to share her facts. One of the things she did that I really liked was tell the student, “Please check your audience to make sure they are ready before you begin.” The student actually stopped, looked, and waited for her peers to be quiet before starting!

Here are the facts the student had written on her board:

  • 5 + 2 = 7
  • 2 + 5 = 7 (When they got to this one, the teacher asked, “What kind of fact is this?” and the students responded that it was a turnaround fact of the previous fact.)
  • 7 – 5 = 2
  • 7 – 2 =5 (Again, the teacher stopped and asked, “What kind of fact is this?” and the students responded that it was a related subtraction fact to the previous fact.)
  • 5 + 2 + 3 = 10
  • 10 = 5 + 2 + 3 (This time the teacher stopped and asked, “This one’s interesting. The 10 is on this side of the equal sign. Why does that work?” She called on a student who gave an unsure response, but he kept at it, and the beautiful thing is that she gave him plenty of think time rather than quickly passing to another student. She knew equality was a tough concept for her students so she wanted to slow down at this point. After the student finally articulated that the values on both sides of the equal sign were equal to 10, she called on another student, “What does the equal sign mean?” and another student, “What does the equal sign mean?” and a third student, “What does the equal sign mean?” I saw that she does this repeated questioning anytime there is an important point that she wants to make sure the class hears multiple times.)

She did the same routine holding up 9 fingers and had another volunteer share his answer. Then she told the students that they were going to count and needed to get in their counting circle. What surprised me is that her students all quickly put away their white boards and ran over to another area of the room to sit in a circle. I’m envious of her fast transitions. Her students wasted no time. She was able to start her counting circle in less than a minute.

She sat down on the floor with them and told them they were going to start by counting to 120 by ones. She said, “One,” passed the ball she had in her hands to the next student, and off they went. The class quickly counted, although a couple of students got hung up here and there. Again, she did a great job of giving the students think time, and it’s clear she taught the other students to be respectful of one another because the other students were very patient about waiting.

After counting to 120 by ones, she had the students count backward by ones starting at 90. She said, “90,” passed off the ball, and then they came to a screeching halt. The first student had no idea what to say! She thought for a minute and finally said, “91?” The teacher replied, “Remember, we’re counting backward, not forward.” She gave the student a little more time and then said, “Do you want to get some help?” The student nodded and then ran over to their class hundred chart. After consulting it, she came back and said, “89,” with confidence. From there the count continued a bit more slowly than counting forward, but still very smoothly. When the count got back to that first girl, she had no trouble at all.

Next the teacher challenged the students by saying they were going to start at 33 and count to 52. They must have been familiar with this routine because the students immediately started trying to figure out who was going to say 52. Once all the students held their thumbs up to indicate they they were ready, she asked for their predictions. It was interesting to see that four different students were guessed, but they were four students all sitting next to each other, so clearly the students were all within a reasonable range, but not perfectly accurate. They tested out the count to see who it would actually land on and then jumped into yet another count! For their final count, they counted forward by tens to 120. Because the count went through so few students, she actually had the count go clockwise and counterclockwise to ensure every student got to take part at least once. Nicely done!

Then the students were up yet again. This time they moved back to the area they were sitting in when class started. They continued counting by tens, but this time they used a Math Rack.

  
 

The students clearly have used this tool countless times throughout the year. They were able to count by tens from 0, from non-zero starting numbers, and they even counted backward by tens from 100, 98, and 91. I liked how she even had them make connections to their work this year making tens.  When the students counted by tens and got to 93, she covered the remaining counters and said, “We’re at 93. How many more to 100?” When a student said, “7,” she replied, “How did you know?” and the student said it was like a tens fact, he knew 3 + 7 is 10.

And that’s not all! She had the students hop up and meet her back on the carpet where they had done the counting circle so she could show them how to play a new game. It was a matching game. Students drew a green  card that would say, “What is 10 more than ___?” The blank was filled in with a number. The students then drew a white card looking for the matching answer. After teaching students the game, they got to play it in pairs for 5-10 minutes.

And that’s not all! When the game was over, the students quickly put away their cards and met the teacher back on the carpet where class began. She started showing them an activity they were going to continue the next day. This time they would play a game by themselves where they would roll a dice that would give them a starting number from 1-6. Then in their journals, they needed to count by tens starting at that number and record their counting pattern in their journal. She modeled one turn and then had other students model a few more turns before the class ended.

Whew!

Recounting it all, I can’t believe how much math these kids did in 60 minutes. It was incredible! And while it seems like they bounced around a lot, it was all very intentional. First of all, she planned today’s activities based on her formative assessment of the students after they did several lessons on counting by tens the previous week. She didn’t feel they were strong enough so she wanted to give them more practice. She also wanted to ensure they were comfortable doing it in multiple ways. This was such a smart move on her part because it would have been easy to think that her students “got it” early in the lesson, but when I played the matching game with one of her students, I was intrigued that the student knew 10 more instantly for some of the cards, but for others she had to consult the hundred chart.

The other thing I loved was how many opportunities she gave the students to get up and move around. It would have been very easy for her to set up shop at the front of the room and try to get through all of her planned activities. However, as we all know, students get very fidgety after sitting for a few minutes. While her students did sit for most of the class period, the constant change in locale got them up and moving and made the class feel very active, energetic, and fun.

And above all else, I was struck at how much respect the teacher showed her students by giving them the think time that they needed and how excited she was to hear their thinking and ideas.  Her respect has clearly rubbed off on her students because I heard numerous unsolicited compliments given out by students whenever a classmate was willing to go to the front to share their work or demonstrate counting by ten. It’s such a great mathematical community, and I’m so happy I got to visit even if just for a little bit.

Fraction Number Sense

This week I gave a presentation to grade 3-5 teachers from around my district. One of the points I wanted to drive home to them is that number sense does not apply to only whole numbers. Students can (and should!) have number sense with regards to fractions and decimals as well.

One area where I see teachers frequently take shortcuts and/or avoid number sense is with comparing fractions. Some teachers teach their students to use cross multiplication to verify whether two fractions are equal. Others teach their students to use one and only one strategy for comparing fractions: find common denominators. This strategy works, don’t get me wrong. However, it develops one skill, not number sense.

When comparing fractions, students should be mentally choosing from a variety of strategies. Why? Because students might notice they can make a comparison quickly and mentally. Why go to the trouble of creating common denominators, which likely involves making some notes on paper, if you can mentally make a comparison based on your understanding of fractions?

To that end, I put together three “books” that teachers could use to prompt some discussion and reasoning among their students. I’d love for their students to start realizing that fractions are something they can make sense of.

Each book focuses on comparisons around a particular strategy, but the strategy is never spelled out for the students. Instead, my hope is that by noticing, wondering, questioning, reasoning, and communicating, a class of students can make sense of the strategy in each book.

By no means are these books intended to fully develop students’ number sense with regards to comparing fractions. Additional experiences and practice are likely required. However, if you’re a teacher who wants a place to start this work with your students, give these a try and let me know how it goes.