Well, every year, since 2017, I get the pleasure of hosting the Build Math Minds Virtual Math Summit. This year, the free summit for elementary educators will be taking place July 29th and 30th, 2021. Registration is now open. 

I’m Christina Tondevold, The Recovering Traditionalist, and today I’m giving you a Virtual Math Summit 2021 preview in our quest to build our math minds so we can build the math minds of our kiddos. 

Watch the video or read the transcript below:

Watch the video or read the transcript below:

Here are links to products/activities mentioned in this vlog.

Register for the 2021 Virtual Math Summit

Theresa Wills’ amazing math templates

I know that after the 2020-2021 school year, you may not want to be thinking about or planning or preparing for next school year. But as I finish out the vlogs for this year, I like to end with a reminder that we have the annual virtual math summit coming up in July. Hopefully you can get that well-deserved break and spend some time doing some of your favorite things this summer. I just hope that maybe one of them might be attending the summit. 

Now, to get you excited, I’m doing things a little bit differently this year. One of our amazing presenters this year is Theresa Wills. Theresa caught my attention this last year and a half, with the coolest templates that help keep your students engaged and doing rigorous mathematics in an online setting.

I know many of you are back to in-person teaching but these templates can be used even in an in-person setting. They are so cool and engaging that I wanted to make sure that it gets shared with everybody. They give you a way to allow all your students to have a voice and show their thinking without needing to be up in the front of the class.

So instead of me talking about them, I got the okay from Theresa to go ahead and share her entire virtual math summit session with you today. I really wanted to get this information out to you before you finish up school because the templates can really help you create that fun and engaging lesson for that last bit of school when everybody is kind of checked out and ready to be done.

Don’t get me wrong, these templates and lessons will be amazing ways to engage your kids throughout the year. But I know that everyone right now is just kind of spent. So I thought that by pre-releasing Theresa’s session, it would help brighten up the end of the school year for you.

Don’t forget to make sure you get registered for the full virtual math summit so that you can watch all 20 sessions that will help you get excited about teaching math next fall. They don’t release until the end of July. This is just a pre-release of Theresa’s session.

Alright, I’m going to hand this over to Theresa.

Teaching Math at a Distance: Routines That Foster Student Voice Presented by Theresa Wills, PhD

Hi everyone, I’m Theresa Wills and welcome to the Virtual Math Summit. 

Quick routines and warmups are efficient ways of practicing reasoning, uncovering prior knowledge and engaging students to participate with open-ended questions and many possible answers. It is the perfect place to begin trying out some different online modalities.

All of these routines that you’re going to see today have been tested out in live online classrooms but they’re not limited to just virtual teaching. They work great in concurrent or hybrid, whether you teach some students virtually on some days or you teach your virtual students and in-person students at the exact same time, these routines work great.

Your in-person students might use a whiteboard and simply show you their thinking as opposed to typing it into the text boxes. But if your in-person students have a device available, they can interact seamlessly. Similarly, your virtual students could use a whiteboard but what I’m going to show you today transforms this so that you don’t have to squint your eyes and see what they see. Instead, their voice is loud and clear on the slides and it gives each and every single student a voice.

Reflect & Reimagine 

We’re going to look at 5 different math routines that you can do in the face-to-face, virtual, hybrid or concurrent classroom. In each routine, there will be a section that says reflect and re-imagine. There are 3 focus questions: 

How do students have a voice?

How does a teacher display student voices and strategies?

What are the benefits of this math routine?

Estimation Routine

So let’s jump right in. We’re going to start off with a routine called estimation. By the way, I love the resources that come from estimation180.com, so I want to throw out a shout out there. There’s already so many images that you can use. Let’s take a look at what this looks like in the online classroom.

I’ll typically start off with a slide that looks just like this. As a classroom management tip, I typically assign one color to a student at the beginning of the year. They have this color all year long. They know exactly where to type and they always look for this color when it’s time to type. Now there is an image in the middle, in this case I took an image from estimation180.com because there’s already so many. My students are filling in a number that is too small, too few marshmallows in that cup.

When I want to get a student’s voice, I’ll move their thinking right over. I kept their original. I just made a copy and moved it over. Now I’m going to ask that student to tell us, “Why did you select 20?” And I’m going to try to type their voice exactly as I heard it in the text box. “I think it’s too low because I can actually count 20. And so I know there’s a lot more than that.”

Next I ask the students in their same color to type an estimate that is too high. If you’re looking for extra accountability, notice that the color coding allows you to see your students’ thinking in terms of too low and too high. So in that bright green one down by the bottom, I could see that their too high is a million. That leads me into thinking of follow-up questions. 

Let’s hear why this student picked 48 marshmallows as being too high. – Yeah, what I did is I took the largest ring at the top and I multiplied that by the stack of marshmallows I found, so that was 8 x 6. 

Now that we have had a chance to hear two different student strategies for why this is too low and why this is too high, we’re going to be able to see what the real estimate is, how many marshmallows are actually here. But this conversation about the too low and too high is just as important as the conversation about the just right. 

Now students are typing in their real estimate for how many marshmallows are in this cup. What is most interesting about this routine is that by doing the too low and the too high beforehand, I find that my students’ range for this actual estimate is tighter. So we don’t have too many students that are saying a million or 500, they’re a little bit more accurate and it’s because they listened to their peer strategies earlier.

Let’s hear why this student thought it was 38.

“I just looked at the top and noticed there were more and on the bottom there were less. So I guessed that there were 5 or 6 and then there’s roughly three layers that might be 5 to 6. And then I went down a little bit, maybe like 5 or 4 and counted how many layers. Then just added all those together and got roughly 30.”

Some final classroom management tips to get the most out of this is that you can record your student thinking by their color code. I could really quickly see where that bright green one was before when they guessed 10 was too low, a million was too high, 37 is just right. And I can know a little bit more about my students because I can assign them a color and know who is which.

Another classroom management tool here is that by dragging their color up, they have a few seconds notice that I would like to call on you. 

Let’s take a moment and revisit those reflect and re-imagine questions. Students know where to type their answer because they’d been assigned a color and they always know where to type just like they always know what spot in the carpet to go on to.

Having that place is the best way for students to really own it. The teacher can display student voices and strategies by dragging over the student box and asking them to elaborate and then typing their voice exactly as heard.

Here’s my protocol for using the estimation routine in an online class. This estimation routine is great for early number sense but also fantastic for calculus, algebra and more.

In this example, I have a vessel that is going to be filled up. Over time, that bottom section will take longer to fill than the top section. So what I’ve asked students to do here is to consider the data from 1 second, 5 seconds and 10 seconds and guess the height full after 20 seconds. 

Clothesline Math

Let’s take a look at clothesline math. When doing clothesline math, I simply start off with a number line and endpoints. However, you can do this even without endpoints if you want an open number line. Then I have my students type their number in and drag them down. Here you’re seeing two slides because I had a really large class that day. Once students type their numbers, they drag them down to the number line and they start to order them. The interesting thing about ordering these numbers on the number line is that it’s not static.

This dynamic representation actually requires the class to work cooperatively to move their thinking around. Once one student moves it, it might affect the way that others need to be moved. Finally, I’ll highlight a couple of the numbers for students to elaborate more over the microphone. 

There’s some great advantages to this routine and the most important one being student agency. So here students are deciding what number they put in. It’s not some worksheet or something where I as the teacher determined the number, instead, they’re able to show me their real understanding of more complex numbers. 

Let’s reflect and re-imagine. The neat thing about this activity is students know exactly where to type their number and then to drag it down. Now, this dragging of the number down helps us to display their thinking about where the number falls between other numbers and the number line.

Here’s my protocol for doing clothesline math. First, the students create it. They create their number. Then they move it. Then they share it. And as more students are adding, they revise their thinking and move that number around. This routine is great for early number sense such as what numbers would fit between 100 and 1,000 but it also works great for rational numbers and integers.

Convince Me That

Let’s check out the ‘convince me that’ routine. This fantastic routine was found off of convincemethat.scmath.org, so shout out to more resources that are already out there. In this routine, every single student gets a slide. Now, yes, that means that my slide deck is already 25 slides long but the reason for this is that every student gets their own space to show their own proof.

I’ll leave some materials off to the side, some on or just keep the slide completely blank. This will allow students to get creative and flexible in their thinking as they convince me of the problem. One advantage of doing ‘convince me that’ in this collaborative space where I can see everybody else’s slides is if I’m ever stuck in thinking, “Gosh, why does this work? 5 divided by half is greater than five? Wait a minute. I thought it was 2.5. That’s not greater than 5. What is this?”

Now I am learning from other people, I’m learning from my peers. I’m not just copying, I’m inferring what they’re doing and when I think I’ve understood it, I’m modeling it on my own slide. So that takes into account interpretation, evaluation, and creativity.

When I do visit a peer slide and I want to show them, “Oh, wow, I connect with that. I got a great idea here.’ I’m going to move an emoji right next to their thinking. So that I can give them a shout out.

“Naomi, can you explain what you were thinking?” – “Yeah, you see how there’s 5 circles but we’re only looking at half of it. So really there’s 10 pieces like there’s 5 pinks and 5, I don’t know, greenish color, and put together, that 5 + 5 is 10 and 10 is greater than 5.”

“Oh, that is a really great way of showing it. And Kathy, it looks like you showed something similar using the number line. Can you explain that, Kathy?” – “Yes, it’s the same thing. There are 10 jumps there. The jumps are like the bounces on the number line.”

“Gail, it looks like you’ve done something a little bit different than the previous two. Can you explain your thinking?” – “Yeah, so, if you’re halfway between 5, it’s 2 and a half but you need to see that it’s not 2 and a half, five is the halfway point. So if 5 is the halfway point, then the whole trail is 10.”

Because each and every student has their own slide, their own space, they know exactly where to put their thinking. The teacher can display the student voices by having the entire class move to the slide that the student is talking about. For example, you saw Gail’s slide about a trail and the halfway point, that was a student voice using student strategies.

The protocol that I used for the ‘convince me that’ routine was first to assign a slide so that every single student gets a voice. Then make sure that the students are able to move text boxes, shapes, scribbles, and more to show their thinking. I love it when students cooperate together and look at other people’s slides for ideas but I also want to make sure that they are giving feedback to their peers, so I leave some emojis on the side. Finally, I select a couple of students to share it. The ‘convince me that routine’ works great. This was an example of a fraction division but this can also be used in the secondary.

For example, in this image of a graph, I said, “Convince me that the Y-intercept is not one.” 

What Could This Be? 

Another fantastic routine that supports students thinking flexibly is the ‘what could this be?’ routine. Here in this classroom, I started off by putting down one 100 flat, three 10 rods and five unit cubes. But what I’m looking for is over 20 different situations of what this could be.

As students start to type in their thinking, I challenge them to not make any duplicates. If somebody else has already typed your thinking, show a connection like this was my idea too by changing their font to bold. Then think of another way. This way, it validates students for having that initial thought and it also gives feedback to their peers.

Next, they’ll use color coding so that students know ahead of time when I plan to call on them and how their representation matches. I’m usually very specific when I select just a couple of students. It’s not random. I’m usually looking for very similar representations so that students can make strong connections between them, or if they are pretty fluent in representations for this, I’ll look at two very different ones so that it stretches their mind to think flexibly about why they’re actually quite similar.

“I was thinking the 100 flat as my unit, my base unit one. And then if that’s the case, then my 10 rods or my rods will become my tenths. And then my single blocks would become like hundredths. So I figured that it would be 1 and 35 hundredths.” 

“Neat, and somebody else had written it out. I just want them to go ahead and connect to yours.” – “The flat would be like representing one whole. Then it would be 1 and 35 hundredths.”

“Awesome. How about the other one that’s green?” – “That’s me. I said it could represent money. It could be $1 and then 35 hundredths or 35 cents.”

In this routine, students know that they can type in any available space. Since many students have the same idea, it’s important that we give them the opportunity to make connections by turning another peer’s thinking bold if you had the same idea. Finally, to get that student voice out there, the teacher selects very purposefully the couple that they’re going to talk about and the whole group and they ask the students to explain their thinking.

The protocol that I use in ‘what could this be?’ is first that we have a class brainstorm, that each student imagines a different way that the model can be interpreted. Then we “agree boldly.” If another student typed your idea, they just turn the font bold. Then highlight notice: this is where the teacher selects and sequences ideas with a highlighting color. The highlighting color will help your student to quickly find their microphone, get their thinking together, and classroom management wise, it will make a more seamless conversation from one idea to the next, to the next and the next. Then those students who are highlighted turn on their mics and they share. The ‘what could this be?’ routine is fantastic for early number sense but it’s also great in secondary. 

For example, this might look like an ordinary parabola but what is it really? What could it be modeling? When would we have something that is more in the negative space than the positive space? What is something special that happened at 5? These are the questions that students will answer.

Which One Doesn’t Belong?

Let’s take a look at how ‘which one doesn’t belong?’ can be modeled three different ways. In the first way, we have students move the doesn’t belong symbol to whichever quadrant they think doesn’t belong based on a rule that they make themselves. I’ll even usually call on a couple of students, but for the most part, this is an anonymous page.

The best part about an anonymous page, especially if your students need more confidence in participating online, is as a shy seventh grader, I can look at this and say, “Oh wow, somebody else has the same idea as me.” That is automatically going to grow my confidence so that in the next part, I’m more willing to participate.

Once we’ve taken a poll of the class on which ones they think don’t belong, now I want to know, why doesn’t it belong? I want an actual rationale. This rationale is really important because it gives me insight not only in my students’ thinking, but it lets me know what vocabulary they know, what connections they’re making, and what ideas did I not even think of? So I’ll call on a couple of these students to share out loud and elaborate a little bit more.

“[Green] Because the sides are yellow, it’s a rectangle. The red is a rectangle and the other ones are all square. Even a cube has square faces. And then that has square pieces on it. But the yellow and blue are definitely rectangles. I think even the red is a rectangle.”

“[Pink] Can you tell us what monochrome means?” – “So this one just only has one color. The rest are colored. That’s only white and gray.”

“[Yellow] Tell us a little bit more about the idea of manipulation.” – “This is the only one. Well, like, you can turn it. So there’s two 3D shapes, but this one you can turn, you can twist, it’s a toy.”

“Well, folks, these are some really great ways of thinking about this. You have now convinced me to maybe change my mind. So we’re going to move down to slide nine and this is our accountability slide. Here is where you’re going to put your name, the letter you think has the best reason for not belonging and your description that you’re going to hold yourself accountable for. You can change your reason, you can pick somebody else’s reason but you’re going to now commit. This is the committing part of the math.”

The beauty of the ‘which one doesn’t belong?’ routine is all in the balance of anonymity. So in those first couple of slides, students knew where to move their doesn’t belong symbol or where to type their reasoning based on the quadrant that they decided. In the last slide, they knew where to put it because any available space was open to them. In terms of teachers displaying student voices, it was the student voices that were heard and seen on every single slide.

The protocol that I used in this routine was first strike it. Day one, we use the strike it symbol and we show which one doesn’t belong as a whole class poll. Day two, I look at giving a rationale, why doesn’t it belong? On day two, I have students use a bold text to show connections. So you saw in the classroom video, when someone agreed with somebody else, they changed their font to bold.

It’s a way of preserving the original thinking while making a connection. Finally, on the accountability slide, students are learning to make their choice public but after some proper instruction such as defending your reasoning and listening to other students’ thoughts. 

‘Which one doesn’t belong?’ is a great resource to use in primary grades, early elementary, upper elementary, middle school, and even secondary. Let’s revisit the quote that I stated at the beginning.

I hope what you gathered from today’s presentation are ways that you can spend just a quick seven minutes of your class introducing one of these routines and getting each and every student the opportunity to share their voice.

As I conclude, I’d love to share a quote with you from Virgin Thompson and this should motivate you to try something not once, but three times. 

I’m Theresa Wills and I look forward to hearing how you use routines in your math class to foster student voice. 

In Conclusion

I don’t know about you but I just love the part of her presentation where you get to see the participation in real time. If you want to go get those templates, don’t forget, use the link that’s underneath the video above and it’ll take you to Theresa’s site so you can download those amazing templates. Also underneath the video is your link to get registered for the entire Virtual Math Summit.

I hope to see you in July for more awesome, informative and engaging sessions just like this one, for the 2021 Virtual Math Summit that is presented by Build Math Minds. You can just go to BuildMathMinds.com/VMS21 to get registered. 

All right, I hope this video, like all the others, helps to build your math mind so you can go build the math minds of your students. Have a great day and I can’t wait to see you at the Virtual Math Summit.

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