About the Authors
Amanda (Mandy) Jansen is a Professor in the School of Education at the University of Delaware where she teaches future elementary and middle school mathematics teachers. Prior to becoming a professor, she taught junior high (grades 7-9) mathematics in Mesa, Arizona. She earned her Ph.D. in educational psychology from Michigan State University.
High school math teacher and debate coach Chris Luzniak longed to see his math students as empowered and engaged as the students on his debate team, so he began incorporating debate structures and techniques into his math classes. Excited by the growth he saw in mathematical thinking and participation, Chris continued experimenting. Over the years, he developed and refined math debate routines that really work, which he uses daily in his current middle and high school classroom. Chris has been teaching math for the past 15 years, beginning in New York City public schools, and now currently at The Archer School in Los Angeles.
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Nate: What's the current state of your state with regard to back to school?
Mandy, in Delaware, if my information is up to date, you have three possible scenarios depending on how things develop in the coming weeks, yes?
Mandy: The governor got together a committee and some working groups and did surveys of students and parents and teachers, and they developed three plans depending on how extensive the cases are at that time. The plans are what it takes to be fully online, what it takes to be hybrid, these various versions. We're told that the state, the governor will come out with a clear stance sometime in early August, so that's for the K-12 system. A lot of us who are teachers or work with teachers, that uncertainty creates another level of stress because folks want to plan, but then also they want to be safe and be healthy.
Mandy: The state, I think, I really appreciated the way they've taken into account a lot of different stakeholders' perspectives when making the plan. Right now, there are guidelines and recommendations to think about. We don't know what phase we're going to be in for the schools, but once we are told about that then you know what the recommendations are from there.
Nate: And you’re a professor at University of Delaware?
Mandy: Yes. I work at the University of Delaware and I teach future teachers, and then I do professional development with practicing teachers. I teach undergrads and the University of Delaware originally had a plan that they we were wanting to be some version of hybrid, where it would be face-to-face one week, and then off campus the other week and alternating which classes were on campus, but just last week, they changed and they said, "We're all going to be online."
Mandy: Now that looks like the fall is going to be online. Originally, it was going to hybrid through Thanksgiving and then online after Thanksgiving, but now it's all online for the fall.
Nate: I wonder if universities are in general, better equipped for online learning compared to K12 schools
Mandy: That's an interesting question. Universities often had already offered a range of classes online before, so there were models in place, but individual faculty were inconsistently prepared and ready, so some folks who have never talk an online class before, it was the same kind of big lift. Then the question about what is more equitable or supportive of students, to be synchronous or asynchronous, was a decision often left to the faculty member to sort out with their students. That was a complicated set of decisions to make as well.
Chris: Are you going to be more synchronous in your classes?
Mandy: I moved my class; I taught a math class for future elementary teachers, and I moved that class online, and I did a lot of surveying of the students, what resources they had, where they were going to be able to access the internet, would it be fast enough for synchronous, would they have printers, what do they have? Based on that, and then the overwhelming student response, "We want a math class to be synchronous. We want a math class to be in real time." Because they seemed to have the wifi access in my class, we went synchronous.
Mandy: I think one reason why it worked is they had some other classes that were asynchronous, so I worry if all of their classes were synchronous, what kind of drain that would be on them, but they seemed to appreciate the real time nature of it.
Nate: And Chris, you teach in Los Angeles County, yes?
Chris: Correct, yes. LA and San Diego just a few weeks ago declared that they're going fully online for the entire fall semester. I'm at an independent school, so we don't have to follow that, but we at least are going to start the year fully online. It's kind of like Mandy was saying, we had a hybrid plan of a week on campus and a week online, but that's now pushed off for some time in the future when we can do that. I think for at least August, September, we're definitely online, and we're going to kind of take it month to month and see how things go.
Chris: It's really hard because our school is so much built on the quirky, fun community building. I'm at an all girls' school, so the relationships and the presence on campus and all the fun, weird things that we do as part of our school culture is such a trademark for the students, it's really hard for them to be separated online.
Mandy: Chris, did it reduce any sort of stress for you when the school said, "We're going to go completely online," or are you missing the notion that you could see them face to face sometimes? How did you feel?
Chris: I think it was, for both myself and many colleagues, it was stress reducing rather than the uncertainty of this back and forth, are we going to open online or are we going to still do a hybrid? It was kind of waiting from day to day. Just having a decision, even if the world suddenly got better tomorrow, just having a decision in place so that now I can plan for a couple weeks from now. It's not that far away until the first day. We can figure out together as a department and as a faculty how to best connect with students online and start their courses and start the year online. It's really helpful, the more advanced notice we get the more I can start to plan ahead, and, I imagine, same for you.
Mandy: I'm actually on Sabbatical next school year, so I'm not teaching. I'm working on research and writing. But I very much am invested in what's happening because my colleagues, teachers I collaborate with, but I would have already been at home.
Chris: Lucky timing.
Nate: Chris, you mentioned about the relationship building that happens at the beginning of the year, how are you and your colleagues, how are you going to do that online?
Chris: Relationships and social-emotional learning and all that stuff is at the forefront of all of our faculty's thinking and definitely all of the questions that come up within our department meetings that we've had. We've had a couple optional department meetings over the summer. Everyone's really concerned, but also constantly thinking about it and coming up with different ideas and sharing with each other.
Chris: I think our first lesson back is going to be a lot more getting to know you and silly, ice-breaker questions and activities and things than I even would normally do when we would be in-person on the first day or two. Our middle school is having two full extra days earlier than the high school so they can have a lot more time ... a lot of those students are brand new to the school, because we're a six through 12 school. A lot of our sixth and seventh graders are new to the school, so they want to have lots of time of just getting connected and building a little bit of relationships and getting to know the teachers and the school and the online routines as well.
Chris: I'm thinking a lot about, like Mindy, you were saying, my math students really did like synchronous. Even just 20 minutes of synchronous, we kind of connect and do some stuff, and then I can send them off to do some problems with a friend virtually or solo. I'm thinking about when I do some synchronous time and do breakout rooms, or giving each breakout room a fun, non-mathy question. I'm thinking also this year some of them won't even know their classmates. When we went online in the spring last year, they already knew each other, but some of them don't know some of their classmates yet. A lot of them don't know me yet. I think in every single class period there's going to be a, "Go to breakout rooms, do the silly question, and then do these three math problems," or something like that.
Mandy: That's actually one of the pieces of feedback that my students gave me at the end of last semester. We're a college class, we're here to do math, and they said, "You know, it could have been really great if you had done a little bit more of the stuff where we could just talk about life," because I was trying to having a rich mathematical experience, but they thought I could do a little more community building. I would normally do some of that face to face, so they had a good point, so I think that's great.
Mandy: Because it's very different, too, to shift online after you've had time face to face versus if you're just meeting them. I think doubling down on that effort will go a long way.
Chris: Yeah. I was just blogging about this the other day. Using my debate structures from my book I think will give kids a little bit of a safety net with people they don't know, of a talking routine of how to say something a little bit personal, so we'll have "little silly debates" I going to call them, for just like two minutes at the start of each breakout.
Mandy: That's awesome.
Nate: Do you think there are aspects of distance learning that can improve student talk about math? I think online learning has, sometimes been portrayed as a poor substitute for in person teaching. But I’m wondering if this mode of communication doesn’t have some strengths…
Mandy: Yeah, my students told me the benefits that they experienced. Just to set a context, the way my math class would typically go would be we would have a problem that would be launched and then they would work on it in groups. They would come share their drafts in progress of their initial thinking in groups. I would select and sequence a particular draft or two to be shared. Then they would go back to their groups and revise some more and then share out how their thinking evolved and changed. That would be how rough drafting goes.
Mandy: How they said it worked differently online was, in class, when you're working in your group, in face to face, you hear the other groups talking, and that can effect either how you think about the math or how you feel about your work on the math because you wonder, should I be further ahead or should I be trying another strategy. But in Zoom, when you're in the breakout groups, we use Zoom as our platform, but it's the same, if you're in a breakout room, you can't hear the other groups. They said they were more focused on just connecting with each other and not thinking about what the other groups were doing.
Mandy: I was also having the groups synchronously put up their evolving drafts on Google Slides and the whole class had access to that slide deck. People could do this with Jamboard in other ways, too. I think next year I'd try Jamboard instead of Google Slides. The point being, Wikispace, where people can put up their drafts, but they could all look at each others' drafts much more easily than face to face. They could see, what's group one doing? Even though they couldn't hear it, if they wanted to look, they could see. I was fine with that. The online space allowed more thinking to be visible to the whole class more quickly.
Mandy: Then, after the groups were at a space where we began to talk about drafts in progress, they could go and look at each others' Google Slides, type notes, so you could hear more voices, not necessarily just the speakers because they could go into the notes section of Google Slides and write what they thought, so more groups got feedback from more people. I would say that those are two types of advantages we found. The students told me these were advantages to them, which I appreciated hearing.
Chris: I would say it was similar. I don't think it added something that strong in my classes. I work at a school where we're a one to one laptop school, so all students are given a laptop. Students are already used to doing things like Desmos activity builders. When they submit their response, their explanation to a problem, it will show on the screen responses from other students who have already gotten to that question. Among the other things I use as well, technology-wise, they're already used to the mix of in-person talking and digitally talking; we just kind of cut out a lot of the in-person talking.
Chris: I talked a lot with teachers lately about, I have discovered a personal rule that in Zoom, no one talks when there's more than four or five people in the room. I have to do breakout rooms if I want them to openly talk. When I have the whole class in front of me, students don't speak up. One or two might, but in general, students don't. I had to adapt a little bit with using the chat box, using polls and stuff, just to get some things going, and then going to breakout rooms.
Chris: I wouldn't say the amount of discord has decreased at all, it's just I had to adapt a little bit to the online space.
Mandy: Were there particular ways that the online space seems to support debating, for you?
Chris: That's the thing I'm working on. I played with it a little bit, but the crisis mode of the spring semester was just kind of getting through things the best I can and creating good experience. This summer, when I have had a few moments to think about it, I've been trying to reflect on things. Someone actually on Twitter shared with me a website that's made for building debates and built this little tree of pros and cons list that I'm going to play with, having groups do. Kind of like you said with Google Slides, they can see what different groups thinking is, as a class we can build this pros and cons list for something of, would you rather use the elimination method or substitution, as the two columns. Yeah. I forgot what your question was.
Mandy: I think that's great. I also think it's really important for us to be real with each other because you're saying what's in progress about your work. You're saying you're moving further in that direction. It's not realistic to expect to move online and to feel like we've already arrived in this space fully formed.
Chris: Yeah. My debates have taken much more of a writing version than a speaking version because it is so hard to speak up in Zoom, unless you're in a breakout room.
Nate: Are you using any other formats aside from Zoom? Are there other services or apps that could complement Zoom’s features?
Mandy: I can tell you one thing I want to do and I actually tried it for a book club with teachers around rough draft math, is Flipgrid is really new to me and I think that that would be a great way for people to share their drafts or do debates and then share how their thinking changed and how their arguments would change. Have you tried Flipgrid yet, Chris?
Chris: Once in the spring, and I liked it a lot. There's actually someone on Twitter now that you remind me of that, who did some debate stuff on Flipgrid and she shared her link with me so I could see it. The kids all made a video of their claim and warrant, what their argument was, and they each had to respond to one or two other students who didn't have a response yet. It was neat.
Mandy: Yeah. I think people could talk about, "Here's my first draft of my thinking with this problem," and then people could share their multiple drafts. Then people could interact with each other about their draft ideas and then they could have a new one where they post, "Now here's my revision."
Mandy: I think it also would feel more personal to interact with a video than typing sometimes, and even just changing it up. Maybe in a video you can have more visuals or different things. It's probably not that different. I just think that whole interactivity with seeing people's faces and hearing people's voices, I wonder how that changes how people experience drafting or debating.
Chris: Mm-hmm (affirmative), that's true. I do like seeing them.
Mandy: In terms of, you were mentioning communicating with students, I had college students and I was using email to communicate with them, and they said, "Mandy, we use text." I set up a Remind.com system that they could opt into, and then in professional development that I participate in this summer, I was a participant, not a facilitator, the facilitator used Remind.com as a texting system where you could ask the teacher questions during class about the problem, but it was really like ask for extra information, and then you could write back to them. I didn't think about using that as a way to communicate during class, as a way to facilitate ... she had a specific structure: "I'm giving you a problem without enough information and as a group, you need to come up with what kinds of information you need and then you can text me in this system." I wonder about pedagogically, ways of using messaging systems during class. It was helpful to be able to have a way to communicate while the groups were going with other groups you weren't visiting. Zoom has ways to do that, too, but there are options.
Chris: Yeah. Definitely, when they were in Zoom rooms, I got lots of chats, like, "Come to our room. Help us," or there's a way to ask the teacher to come by.
Nate: Have the events of the past six months changed your perspective about education and how you teach?
Chris: That's a great question. The first thing that come to mind when you asked that actually is some of my colleagues in my math department, we had a long talk about what teaching is, or what our goals are. A lot of my colleagues felt whenever their class period "met," they had to be on. They had to be on the computer, looking at kids, interacting in some way. They're so used to the physical bell schedule, and the minute the bell rings they have to be teaching kids math and interacting with students in the classroom. They were having a hard time imagining going asynchronous or any kind of mix of synchronous/asynchronous.
Chris: We talked about rather than your job being physically present with the kids during these select time periods, your job is by Friday will they have learned these two or three key concepts and how do you help them get there, and how do you check in to make sure they're on their way there or they have achieved it. That can be completely asynchronous and you can actually be doing something else, like grading or something, during your class meeting time.
Chris: For math teachers, it was really hard. They're so used to just being on with the students. It was an interesting conversation to talk about in that small way of rethinking what teaching is.
Mandy: I would say that everything and everyone became more vulnerable in this space of teaching and learning, and students shared much more about what's going on in their lives outside of school than I've had access to in the past. I don't know if that's because of what's going on the world, or because I was asking them more.
Mandy: I normally do things like I have meetings with all of my students, I don't have that many of them, so that's easier to do, at the start of the semester and different things that a lot of people do, like the name tents, where students would write a message to you at the end of class and you write back to them for a while that Sara VanDerWerf has taught us about. Google Slides, at the beginning of the semester that Howie [inaudible 00:25:40] taught us about, where students would post a picture and introduce themselves, and you have then a class Google Slide deck.
Mandy: I did all of these things that I normally do, and they became so much more important to have had in place than ever because then they had these relationships with each other and me. But I learned things like student got a new dog, stayed after class to show me things in their room, like the plants that they were growing, and I'm getting pictures now from students about how tall their sunflowers are that have grown, I'm getting texts about new music. I don't normally have that closeness with my students. College students, you only have them one semester, I tend to have a little bit of a higher boundary with personal life, but this is just, we're in a different space.
Mandy: That's just been really lovely and it makes me think maybe I shouldn't always have such high boundaries with my students. At the end of every class, I would always give a Google Form to share something class-related, but then also a very open, "Anything else you want me to know about your lives." I learned about whose family members were impacted by COVID, what kinds of jobs they had or their family members had and how that was affecting their mental state. That helped me just think about how important that is again. Validating things we already know, but really trying to understand the people in the room and the lives that they're living. I was glad that I was able to work with some folks who were willing to be open with me about their lives.
Chris: That's awesome. How often did you say you did those?
Mandy: My class met twice a week. We met Tuesdays/Thursdays in the afternoon for 75 minutes. It was at the end of class, a Google Form that we would do. It would just have usually two questions, something related to class, like, "What is a way that your thinking changed today? What helped that change? What else do you want me to know about what's going on in your life?," or different kinds of prompts like that. Sometimes I would start the next class then with some quotes from the class-related question from the Google Form.
Mandy: I don't think you would need to do them every day. You could do them once or twice a week, or you can do that asynchronously as well.
Chris: I like that. Just the regularity of it as a really short survey.
Mandy: Mm-hmm (affirmative).
Chris: Mandy, our whole world is a rough draft right now.
Mandy: That's exactly right. I often will, when I'm talking with folks about rough drafting, I say, "I didn't realize this, but delving into the concept of rough drafts in learning actually was very liberating for me in my own life as a person." If I'm going to really take seriously the notion that wherever you're at has a strong that we can learn about a build from, then how do I apply that to myself as a person and in my work? Working in spaces that we haven't had to think about and navigate in quite the same way before, so it's a time of reinvention. Just being open to constant revision, I think is hard, and putting ideas out there that are brand new is a vulnerable thing to do, but we're all in that space right now, for sure.
Chris: Yeah. And I would add, I'm trying to teach students to debate that their opinion matters, as long as they can back it up with something. Don't just say anything you want, but say something that you believe in with some reason behind it. I want to hear their opinions on different things. Especially as when summer hit and all the protests that have increased out there, and all the Black Lives Matter, the attention on these social justice causes, I'm hoping to continue this debate stuff going forward to keep empowering the young people, the next generation, so that they can speak up and can keep making a difference.
Mandy: I think we both value to really help people have them have spaces to exercise their voice, and also learning to listen to each other. To engage in debating and to engage in revising, you're not only putting ideas out there, you're trying to be open to other perspectives and to keep learning. I think both of those perspectives are important right now in the context of working for justice and working for peace.
Chris: Yeah. I think we both, in our work, stress that it's okay to be wrong. It's okay to just put something out there and get feedback and then revise. Same with the debates, yeah.
Nate: That’s a great point and it’s a strong theme in both your books and in Stenhouse’s math books….it’s about removing the anxiety around this binary kind of thinking: you’re either right or wrong …
Mandy: Right. I really would love math classes to be a space were students don't feel the pressure to perform being correct all the time, because then there's just only really one way to be mathematically smart. But if you can put out a draft of what you're currently thinking and then people are oriented to try to understand you, trying to make sense out of what you're saying rather than to evaluate you, and trying to see what you can learn from someone else's draft, then more people can have the space to be recognized for their mathematical competence and their strengths. We all want to be heard, so if we can work on trying to hear each other and value what people have to say, I think that's a better environment for everybody, math or otherwise.
Mandy: Also, just normalizing that changing your mind in the face of new information, like initially, we were encouraged not to purchase masks because the healthcare industry needed them. Then, with new information that cloth masks actually help us, we can revise our stance about masks in the face of new information. The idea that rough drafting and revising can be an orientation that helps us with math, but also just as being a citizen in the world I think is important.
Chris: I just ditto all of that. Yes, I agree. I say all the time, "I don't want math class to be about answer getting. I want it to be about sharing your thinking and your processes," and in the debates we're sharing our opinion of what the best method is, or the best model for this situation, and you may be correct in finding a solution, but so is maybe someone else, who does it a totally different way and understanding and embracing that, listening, having empathy, all of that, are life skills that I want the next generations to have.
Nate: Just wondering, does being online perhaps encourage . encourage some students to open up a little bit more?
Mandy: I would say sometimes you hear from different people. Some students who were in the physical space of the face-to-face classroom were not necessarily the ones that were ready to type right away in the chat box or willing to unmute and share. I don't know why different people are drawn to different modalities, but I think they are, and I think that if you have these interactive Google Slides, more people can participate in the same time, or in the chat box or in the poll, then you can have those polls and things in class like Kahoot and different things, but being able to make sure you can have them online seems pretty important to bring everybody in, but I do think certain people are participating in different ways, and I don't know enough about that. Did you see that at all, Chris?
Chris: Yeah, it really varied. There could be some students who were less talkative in some form, in person, who were interested in more chatting or even just speaking on Zoom, but also vice versa. Some kids were less interested in sharing out in the virtual world. I don't have any data or things on it, I just saw a wide change for some students. Some of them were the same.
Mandy: Yeah, and there was this space where a hurdle was if you are working on paper to draw a representation of something, you can normally just turn your paper to show your colleagues, so we had this whole learning curve of, "How do I take a picture of it and upload it to the Google Slides?," and they were all helping each other with that. It became this community project where they wanted to make sure they could show each other's work, so they did have this desire to want to still show that work, which seemed to carry over. After the initial learning curve, it was fine, but being able to put on the Zoom whiteboard, people were more comfortable writing on that whiteboard than others. Different people were more interested in uploading a picture of their work or hanging back and looking at somebody else's picture.
Chris: That's the one thing, when you were talking about how groups would overhear other groups in the classroom and it might give them a little nudge or a little pacing idea, same thing with in groups. It was so easy to just write something down and show it to someone else, and now it's so hard. Without having these other apps which we might invest in, I wish Zoom had a way for a second device to hover over it, like a document camera, hover your phone over something and people could see that screen as well because the quick sharing is what's missing in the virtual world.
Mandy: Yeah. I could sign in also with my iPad and then share the screen on the iPad and write on it and draw on it, but different people have different devices, so not everybody could log in with two devices.
Chris: I don't know if I want 30 kids to each have two devices signed on, and then we have [inaudible].
Mandy: That's a lot to manage.
Chris: Yeah, but that casualness of just being able to show something really quickly is missing. Also, they don't have dry erase boards. I'm a big believer in the vertical, non-permanent surface research, Peter Liljedahl, and I have whiteboards on most of my wall space in my classroom, so students can easily, I send the whole class up to the board with a partner and they can work on something, and they're so used to just trying something and erasing it with their hand, they can quickly keep changing their mind, and it's a lot harder to do that on paper and when you're not even with someone physically.
Chris: The number one thing I learned, also, from online last spring is everything takes at least twice as long, so I do a lot less amount of work.
Mandy: I agree with that. Just taking time to try to understand each other through the video medium, and, like I was saying, putting up your work to be shared, it's extra time, for sure. But that's not a bad thing, because then we think about what is the truly important set of ideas we want the folks to take away, and maybe face-to-face learning actually should be taking that long, too, is one thing I've thought about as well.
Chris: Same here. Yeah. We wonder a lot about why we teach so many tangential topics related to everything or why we have to cover every single nitpicky thing other than the expectation of society or something. Narrow down what really is necessary and I want to build good thinkers and not just people who have these 100 formulas memorized.
Nate: Cool. We’re running low on time, but is there anything you’d like to add?
Mandy: Well, I am definitely interested in hearing from other folks who have tried to move any of these ideas about debate math or rough draft math online themselves, and Chris and I are active on Twitter. I'm MandyMathEd. What's your handle, Chris?
Chris: Just C Luzniak. C and my last name.
Mandy: Yeah, so continuing that conversation that way, folks have shared with me some strategies that they've used. The Google Slides idea is something that I learned from Theresa Wills, who works at George Mason University, and I know a lot of other people who have been using Jamboard and different things. I also have been thinking about how different Google Slides versus Jamboard can be. Jamboard is another interactive slide that everybody can write on at the same time, but the way it's set up for the student is that all these icons are right on the side, like a Post-It and a pen and a text box thing and all this. Google Slides, it's kind of hidden, you have to drag it down, and it just felt more of this interactive space. I wonder what people think about the difference between Jamboard and Google Slides, and the pros and cons of that.
Mandy: In the Google Slides, all the slides are together, but with the Jamboard you have to have all of the groups' Jamboard is kind of open at the same time to compare and contrast them.
Chris: So, give and take, yeah.
Nate: One of the interesting things over the last several months has been the sense that because it’s so up in the air that it's kind of an opportunity to be able to try different things, because the circumstances have changed so much...
Mandy: Right. I think an example of that is playing with folks that have been, if we've been used to teaching synchronously, what does any of this look like in the asynchronous space. I think a lot of us have thought about that in different ways. With rough drafting, it might be there's a problem that it's going to be related to an important idea, but you would have students work on it over multiple days. Whatever your discussion board space is, like in Schoology or something, or Google Classroom, posting a first draft the first couple days of the week, where people comment on it during the week and then you can post a revision, or turn in an individual revision near the end of the week after you've gotten some feedback and seen other people's drafts. I think for me that's a really new thing. What does any of this look like in the asynchronous space?
Mandy: It sounds like, Chris, one of the things you've done would be Desmos in an asynchronous way.
Chris: Yeah, and other things as well, yeah. Back to your comment earlier with respect to Twitter, yes, you and I are out there. We're all entering really uncharted territory of how to move our math classes, math lessons into this virtual world in a large way, even if it's just for a couple months and then the world gets better. Who knows? The community that's on Twitter and relying on each other, because I don't have this figured out and most of my colleagues are struggling in different ways, but the more we share and throw out things, the more we can figure out what works best for us and our students.
Chris: I know even Stenhouse has done a good job keeping the authors connected, like you and I are here, and just having different communities where we can just brainstorm out loud and share ideas. Like, I don't know what Jamboard is. Now I'm going to go look it up.
Mandy: Oh, cool. I agree. It was really helpful last semester when people were sharing about how they were navigating some of the experiences they were having. Even our feelings about it, too, and being a part of this larger process together, there's an aspect of self-compassion is being a part of a common humanity, so being in that space online together through Twitter I think has helped me a lot.
Chris: Yes, for sure. Even just someone complaining about how hard it has been online, which it's nice to know that I'm not alone.
Mandy: You're not alone.
Chris: I empathize with you. Yes, thank you for putting that out there and being vulnerable.