INTRO Aviva Levin: Welcome to Lesson: Impossible, an exploration of educational innovation. I'm your host Aviva Levin. As always, I'm chatting with educators of all types who are on the forefront of pedagogy or making effective changes to old practices. Your lesson, should you choose to accept it, is to explore how to increase engagement, lessen anxiety, and create real-world connections by teaching math through a Conceptual Based Instructional Model.  The special agent assigned to help you with this task is JoAnna Castellano of New Brunswick, New Jersey.

INTRO Aviva Levin: I don’t know about your experience with math, but for me, as a very average math student, it was often, paradoxically, a respite from thinking.  I’d get into class, take out a fresh sheet of paper, and for fifteen to twenty minutes I would faithfully copy notes as my teacher wrote them on the overhead projector.  We might do a few practice questions as a class, and then we were instructed to turn to textbook page whatever and do either the even questions, the odd questions, or on a tough day, all the questions.  At that point I would dig out my discman with my very angsty Fiona Apple CD inside, and dutifully complete the assigned task on my own, plonking the numbers into the equations that I just learned, entering a fugue-like state that would be interrupted by the ringing of a bell.  This episode’s guest, JoAnna Castellano, wants to disrupt this traditional pattern, from the note-taking, to the silent students working individually, to the numbers themselves being divorced from any context.  JoAnna and I spoke in early May over zencastr.

Aviva Levin: So first of all, of course, thank you so much for taking the time to talk to me today. If you don't mind just introducing yourself: who you are what you do.

JoAnna Castellano: I'd really appreciate that as well. Sure. My name is JoAnna Castellano. I'm a mathematics specialist at New Brunswick Middle School in New Brunswick, New Jersey. I am a math instructional coach slash teacher who supports approximately twelve hundred students and support about twenty seven math teachers.

Aviva Levin: Wow. And how did you end up where you are? What was your path to teaching?

JoAnna Castellano: I think I have an interesting path to teaching in general. I originally started out in the private financial sector. I worked there for five years. I actually worked at 7 World Trade Center on 9/11. Unfortunately, there that day. Really was impacted by obviously what I went through. And as I was obtaining my MBA in finance, someone had suggested teaching. I attempted to teach a class. I had to do it in front of three people for a teaching fellow position in New York City. And I passed, was approved, an accepted part of the program. Once I stepped into a classroom, I never looked back and gave my two weeks notice.

Aviva Levin: I'm so sorry to hear about what happened to you, but it obviously led you to a place where you seemed to really love what you're doing.

JoAnna Castellano: Absolutely.

Aviva Levin: So I've been doing some reading of... You've written some articles around the math programs that you do at New Brunswick. So I was wondering if you could just describe what your personal pedagogical perspective is and what your aim is for your school math program?

JoAnna Castellano: My goal is to allow students to take a lead in the classroom. I feel that there should be productive struggle in a math classroom. I even in the beginning of my teaching career, I was very procedural. Step one, this is what to do. Step Two this is what you do. And I wanted students to memorize steps. Obviously, that doesn't work. There is much evidence and research behind the fact that students have to learn math conceptually. So in my classroom and in others' classes, I would like to have students take a lead and a teacher act more as a facilitator.

Aviva Levin: And how do you do that?

JoAnna Castellano: Well, I think it's a collaborative process between the teacher and the students. I believe in promoting high level tasks or high cognitive problems that allow our students to work together, either in pairs or in small groups. Make that task a real world situation, something that they can relate to, and then give them an opportunity to explore and expand on their own together. And then obviously bring it back together. And as a facilitator, I want the students to showcase what it is that they learned through the process.

Aviva Levin: Awesome. So is there an example of a unit or a lesson that you could kind of walk me through that has all of those aspects?

JoAnna Castellano: Recently, I promoted a high level task dealing with Pythagorean Theorem. For non math geeks that's dealing with right triangles. And what I'd like to do is I'd like to incorporate sports as much as I can into a math lesson. Math is everywhere, but especially in sports and people or young  people in a middle school relate to the idea of becoming a baseball star, an NFL star. So there is a particular high level task talking about an AFC football championship where it was the New England Patriots versus, I believe, the Denver Broncos. And it's about a player who intercepted from Tom Brady, ran about ninety-eight yards down the field. And then another player from the other team obviously tackled that player. And it's just exciting. So I introduce it with the video. The students get excited, you know. And then slowly introduce the math concept or really allow the students to explore that math concept by using an exciting video to get them pumped up to get ready to try and use the math concept that I want them to explore. And then once they're exploring, it, is the teacher kind of just guiding the conversation.

Aviva Levin: So asking them to notice certain things and pick up on what they've previously learned to apply that to this new thing. Or there is a teacher completely stepping back and just letting them figure it out?

JoAnna Castellano: Well, I'd like to believe it's a combination of the two. I want students to be able to access relevant knowledge that they already have and try to find the new knowledge or the new math concept within. So I would as a teacher, I would want to facilitate and move from group to group as students are working on a task and asking them assessing and advancing questions. So if a student is struggling, I would want to assess where their understanding is or a misunderstanding is and bring them to the next level without giving them an answer. And the student who is getting the concept, I want to advance them to the essential understanding or the math concept that we're trying to reach in that task.

Aviva Levin: Once the students have then understood the concept and applied the concept. I remember from my own high school math, that was just kind of it. Like, you will now use this task to show your proficiency on a test and maybe next year if it applies. How you how are you able to then keep it going in a way that isn't just regurgitation for a test?

JoAnna Castellano: So that is probably the toughest part, is in making the connection or helping the students make the connection so they know where this math concept is relevant in the real world and where they can apply it. So I usually will ask students to showcase the work that they've put together in their groups. And we call this the shared, discussed, analyzed phase of the conceptual based model. This is the opportunity for the students to share what they've explored, how they expanded their thinking, and how they're connecting it to the essential understanding. And even when there are times that there are errors, we could use that as an opportunity for error analysis. So I'm very careful to never use the word correct or wrong. There's so much math anxiety in the world as it is. I want the students to feel that no matter what they say or do it's a learning opportunity.

Aviva Levin: With this model, which obviously involves a lot of higher order thinking how are you able to differentiate and adapt for students on both ends? Both your struggling learners and those who pick things up incredibly quickly?

JoAnna Castellano: Absolutely. So throughout the lesson, we may have prepared questions on index cards so that when we see a group of students are really getting into the task in there and they're moving a little bit faster maybe than another group, I may already have an advancing question helping them get to that math, essential understanding that I could leave at their desks so that that I can move on to another group. So there they're are ahead. We leave that question. We raise that question. Make sure they understand that question. And another allows me to walk away to work with other groups. But at the same time, not stopping those who are ahead of the game, just the same as those who are struggling. I can stay with those groups to ask additional assessing questions, to maybe clarify what the task is asking, what things they can do, and usually relate it to a real world experience.

Aviva Levin: Would you be willing to kind of walk me through how you go from the germ of the idea of the lesson into the actual lesson plan. So I'm imagining the first step is you decide what the concept is going to be. If I'm your teacher or one of your teachers and I come to you and say, OK, I want to teach my kids ratio and proportions. You say, great. We sit down. And what's our first step?

JoAnna Castellano: So we will discuss what the students have already been engaged with. What the enduring understanding of the lesson is and what essential questions we can ask the students to be sure that they know the enduring understanding, the idea that the teacher wants the students to walk away with.

Aviva Levin: So for ratio and proportions, what would be a likely enduring understanding?

JoAnna Castellano: It could be something as simple as what a ratio is, how a ratio could be represented. But normally, when I'm involved, I'd like to use real world examples so that the students know how this is connecting in the real world.

Aviva Levin: My first thought when I'm thinking of ratio and proportions for me would be if, let's say I've got a sewing pattern and I want to size it up or down for the person that I may be sewing something for. Would that be considered a good real world connection or is it too basic?

JoAnna Castellano: No, that's not basic at all. That incorporates ratio and proportions. That's more of something in math we call scale factoring.

Aviva Levin: First, I should ask, what would be your real, real world example? Because we've got, I'm assuming, very different perspectives. What was the one that came to your mind?

JoAnna Castellano: So something I was thinking of is the Mayans in Mexico. There is a great example, I believe, on illustrative mathematics where they ask students to take a look at a particular Mayan ruin. And they give we give information about the steps of the ruin and possibly how many steps lead to the top of the ruin, what would be the depth of each step? And I tried to include as many pictures as I can because if the students have never been to Mexico or they don't know what a ruin is. You know, that also opens up the task where I grab the students attention, because usually with young people, you want to grab them in the first couple of minutes or I don't have them. So usually I'd like to start with some sort of story. I happened to have recently been to Mexico in the last year, so I would start that. That was a surprise present for my husband. We went to see the Mayan ruins and then I'd get into the story and then I would show them photos and then start talking about the steps. And now I feel like I have captured them and now they want to know more.

Aviva Levin: At any point in the beginning of the story, do you mention the words ratio or proportion? Or is it just about the story at first? And that stuff comes later?

JoAnna Castellano: That's such a great question, because that would give away exactly what I want the students to tell me. So, no, I do not use the word ratio. I do not use the word proportion. It could be something as simple as how many meters above the ground will you be on the 90th step or the 15th step, you know, from the base of the Mayan ruins. So it will never mention the math concept.

Aviva Levin: And is that kind of the big reveal at the end?

JoAnna Castellano: That's what we're hoping to get to. Sure. Along the way, it would be safe for myself or the teacher in the room to if the student comes up with the concept of a ratio or we're well into the concept of ratios and proportions. And the student says, well, the ratio of the steps is this to this. And then we'll be. And then I would be OK. So we're dealing with ratios. What can you tell me a little bit more? Can you expand on that idea? I never want to ask a yes or no question. And I never want to answer a student's question. I want to keep it as this conversation of assessing and advancing the student, but constantly with them giving need more information. And I'm just facilitating.

Aviva Levin: So they've heard this story. They're super jealous of your trip. And you've started asking them questions about the steps based on information that you're giving them. Is this a whole class conversation or is this when it breaks into the smaller groups?

JoAnna Castellano: Exactly. So I'd like to start as a whole group. Make sure that the context is very clear so that if students have any question about the context of the problem. Again, if they didn't know what a ruin was, we could have that conversation, talk about the story. And then I would give them an opportunity individually to work for maybe two to three minutes, gauge where each student is at. So quickly making myself, you know, making my way through the room. And then pairing up or grouping students who I feel that their answers are either very much in sync or very different to help them start the conversation of finishing the problem.

Aviva Levin: And then once everyone has understood or discovered the ratio and proportion essential understanding that we want them to grasp, do you come back as a full group?

JoAnna Castellano: So we definitely come to back to a whole group. But that might not mean that every student got it. So it's very important as the students are going through their work. Again, completely assessing each group or each student. So that I can carefully select and sequence students responses because I wouldn't want to select a student who got everything in the time working together with their partners or groups and have them come explain it first because of the student who really didn't get it. That's going to be above their head. And I don't want them to miss out. So I want to take work from students who might have gotten the very basic concepts of just what the ratio is and let them start. And I select them to talk to the group, the whole group first. And then I would select somebody else who could expand on that idea. Ultimately working maybe to four to five groups. And the fifth student or the fifth group of students can then explain the enduring understandings that I was looking for in this particular task.

Aviva Levin: At that point, and presumably everybody's like, yes, we got that. Is it turning to textbook page 76 and doing the odd numbered questions? What is the follow up once the understanding is there?

JoAnna Castellano: So, no, I don't want them to open up to any page. And actually, my district has moved to not purchasing books at all for the upcoming year, especially since we've been working virtually. The curriculum is available online, but that is not what we want them to do, because I still want them to make the connection. So I want them to reflect. So either I'll give them something, what we call an exit ticket, or cool down, and see if the student truly grasped the enduring understanding before I do move on.

Aviva Levin:  The practice before the assessment is that more of Mayan temples are sewing patterns? Like are you getting bringing in very similar examples to to what you started with?

JoAnna Castellano: Absolutely. So in this particular case, I wanted to bring context into the math situation, but sometimes we can just give them basic numbers, should work with the math concept. But in that, I also want to bring other context. So maybe with ratios, it might be making a fruit punch recipe if you're using orange juice and fruit punch. It could be if hippos eat pumpkins and they eat five pumpkins among every three hippos. I still want to bring different real world situations into those practice problems. So, again, when they think of ratio and proportion, I don't want them just to think of the steps of the Mayan ruin. I want them to think of all the different real world examples that we could bring into play.

Aviva Levin: And then on the assessment, if you're still doing like a traditional closed book assessment is the assessment question hippos and pumpkins?

JoAnna Castellano: First of all, the assessment would be very short. I don't believe in 20 questions, 10 of the multiple choice. I'm really for the student being able to explain their work, to model their mathematics and to use precision. So I would give maybe three to five assessing type of questions where I know they're really math rich. And if the student can answer those, then I know that they've been able to grasp the enduring understandings and all the questions I've been asking them throughout the unit.

Aviva Levin: And you've kind of alluded to this, but what would you say has been your biggest success when it comes to this model and what has been your biggest struggle?

JoAnna Castellano: Wow. So when it comes to area of success, I feel over the past three to four years working on this. My number one success, I would say, is eliminating student math anxiety, building their confidence in the math classroom, because I knew as a young person I struggled with mathematics. I unfortunately failed high school mathematics. I knew what it felt like to sit in the room and be horrified when the teacher might possibly call on me. I would purposely sit behind the tallest person. I am very much in an understanding with my students about that math anxiety. So that's my number one success. Another success is having the teachers buy in. The teachers are trying this now in their classes. They are seeing the success and they're willing to open their classroom doors. That would be one of maybe my biggest struggles at first with this concept is that no one wanted to open their classroom door to me. But through modeling, through showing research, slowly having teachers buy into the work. They have now tried it, have seen their small successes, and I ask them just to take small steps of change. It's not a process that's going to happen overnight. You know, it's going to take time. But once we see it, we will see that when students are learning math conceptually, they are truly understanding what it is that they have to express back. It makes them better problem solvers rather than just memorizing an algorithm or following a procedural manner.

Aviva Levin: Are you able to transfer that conceptual model to an e-learning model or have you had to rely on more old fashioned math teaching through this Covid pause?

JoAnna Castellano: You know, that's a great question. We have, as the district tried to start, which we're now calling a virtual conceptual based model, where we are trying to do this through virtual learning, where just as we would in a class introduce a task, we would in a Google Meet, ask the students to read aloud the task, make sure they understand the context of the task, what the task is asking of them. So at my middle school, we tend to do that on a Monday morning or afternoon. We open up that task. We allow the students to work together individually. There's teacher hours where students can ask questions. And during Monday and Tuesday, the students would return the work. And as teachers we're writing lots of feedback. But some of that feedback is in the form of those assessing and advancing questions to make the student push a little bit more or reach these central understandings. So we're doing that in Google Forms. And then on Wednesday, we're asking the students to come back after the teacher has selected some of the student work to use to show to all the students in the Google Meet and eventually bring everybody to that essential understanding or understandings through that math conversation and accountable talk.

Aviva Levin: I'll definitely put a link to an article that you've written. But if teachers are looking to explore and expand how they're teaching math and this is intriguing to them, what are places that you recommend that they check out? And I can definitely put links to those as well.

JoAnna Castellano: Sure. So I'm a big fan of Dan Meyer, Jo Boaler, Graham Fletcher, Robert Koplinski. These are people who I've seen in person, who I've met, who I've read their books. I follow their work. They're very inspirational to me. I also follow the Institute for Learning at the University of Pittsburgh. I've been fortunate to be part of their PD through my school and my district. They offer amazing professional development on the conceptual based model and with assessing and advancing questions. So a combination of all of those would be a great start.

Aviva Levin: And so my last question and my favorite question to ask is if I give you unlimited funds, unlimited time, full control, what would be your ideal classroom, school or curriculum?

JoAnna Castellano: Wow, that's a great question again. And that's a tough one to answer, because I would love to say a little bit of everything. I was fortunate enough to be part of something called P-Tech. That's something new to New Brunswick that I was able to do an algebra bridge program for 40 students, and that is pathways in technology or in early college, high school. And it gives students an opportunity to complete their high school credits. But at the same time, with a local community college, they are gaining college credit. And the setup is in one classroom we have workbenches where we're standing. We're working with engineering. And then in another room, there's these really comfortable chairs where we're just having open discussion about whether it's math, L.A,. engineering, a little bit of everything. It also has the regular classroom where there are desks and a projector. I think that has been just a beautiful portrayal of a little bit of everything that I would want a flexible seating, allow students to take lead in the classroom, allow them to get up and walk around, because sitting all day is just so limiting. So I would want to give as many tools to spark the students interest, but allow them to get up and move freely and allow for those conversations to take place throughout the day.

Aviva Levin: That sounds like a really cool model, really, for any subject to just have those multi-modal spaces and just. I find too like physically moving from space to space can really trigger something in your brain to change the way that you think or look at something. Like even in the classroom, when you designate certain corners of the classroom for different types of thinking the act of physically moving there can make such a big difference. So to have that over multiple classrooms would be so cool. I love that model.

JoAnna Castellano: Thank you. It is very exciting. And even incorporating a math gallery walk right now, virtually. Having students do escape rooms, you know, just moving virtually within different areas on the Internet since we are not in a physical space. Math movement is so important. Someone I follow on Twitter and I've been able to see in person Sarah Vanderwerth, who does a lot of math movement. It's just fascinating. She comes up with these fantastic activities, one being stand and talks where you give the students something to talk about, where they have to talk for two minutes straight. That doesn't sound long, but for students, that seems very long and they cannot stop talking. So the first student has to talk for the minute and talk about everything that they are observing, they are learning. And then the second student can repeat back what the first student said and then talk about their learning and what they're experiencing. And then they can share out whole group afterwards. So I think the math movement, the math conversation, things like that, is what is empowering our students to become these critical problem thinkers. They're going to be doing jobs that don't even exist today when they graduate high school. So it's very empowering, very inspiring symbol.

Aviva Levin: Thank you so much again for sharing this with me and sharing this with the listener. I really appreciate it. And I, I definitely. Math has not necessarily been my teaching forte, but I've learned a lot. And I think a lot of just in general, what you're talking about applies to all subjects of teaching. So thank you for sharing.

JoAnna Castellano: Thank you for having me. And I agree. I think this can work in any content area. Obviously, I'm a math geek, so I get very excited when it comes to the math world. So I really, truly appreciate you having me and allowing me to talk about this fun work that we're doing.

OUTRO Aviva Levin: So there you have in, JoAnna Castellano on why facilitating conversations about real world math can help students feel more confident and become better problem solvers. If you want to find out more about what innovative educators are doing around the world, go to Lesson: Impossible dot com. And if you like the podcast, please consider forwarding it to your colleagues and rating and reviewing it on iTunes. This has been Lesson: Impossible and I was your host Aviva Levin.