 I'm going to be talking about the PAL process. And the PAL process is what Kathy, actually, this is such a lovely graphic and much nicer than ours. So really, it's quite nice. But the same message is there. And we developed this curriculum planning process from work with high school teachers, actually, in a city a little bit north of us. And the reason we developed it is to really concretize the UDL guidelines so that teachers had a framework. So this morning, we looked at the guidelines from a curriculum purchaser perspective. This afternoon, we're going to look at the guidelines from a lesson unit of study perspective. And what we have with the guidelines, oops, wrong thing. Sorry, sorry. What we have with the guidelines is we want to start with a problem that you want to solve. So if you have lessons or units in which students are successful, you just kind of pat yourself on the back and you move on. And when I say the word successful, what we mean is really that they have achieved the goals, as Kathy talked about. That means that the learning experience was a successful one. The goals were clear, and all students were able to succeed in it. So I always think it's a good idea to pause, think about why that was effective, and then move on. But the lessons you want to start with, because teachers will often say, where do I begin? And so what the advice is, is to begin something that's going to happen in the near future. School is starting in September. You might start thinking about, from a teacher's point of view, what am I going to be teaching in September that I can already predict that some students will not be able to achieve the goals? Those are the problems that you want to solve. The lessons where you can, and you can, you can just predict. My kids won't get factors. My kids won't get this or that. Start there. And then when you start there, you start by saying, OK, what's the, what are my goals? What's the context? What's the learning experience? All the same kind of thing. And this is a recursive, and we'll go through this just a little bit. Then you analyze what are the current methods, materials, and assessments that you're using. Then you think about what are the potential barriers. And from the barriers, you look at the UDL guidelines. You teach the lesson. You pause, reflect it. All my students achieve the goals. That's what it was. And the way you figure that out, of course, is that you look at your assessment. Very simple process, one that's been reinforced by what Kathy has said. So the UDL curriculum framework, Planning for All Learners, and you've heard me say this before, the goal is to minimize barriers, maximize opportunities. That's what we're doing so that all students can achieve the goals. When we talk about curriculum, and we're actually talking about the use of that word right now, but right now we do talk about curriculum. We talk about curriculum being goals, methods, materials, and assessment. Learning experience is also a very nice way to talk about it. Instructional practices, people have talked about maybe using that term, but for today we're using the term curriculum, but we're considering changing that. Because oftentimes, schools think about curriculum as they associate that with scope and sequence, and we really are moving beyond that, but to really take into account the instructional goals, methods, materials, and assessment. Oh, this works, keep forgetting. Okay, so the goal is the starting place. And it's really quite important. You have to clarify your goal. So what we might say is when you're identifying your goal, be aware if the means is embedded in the goal. And so if the how is embedded in the goal, it doesn't mean it's a bad goal, it just means you have to be aware of that. Because if you embed the means in the goal, what happens is you're providing one way to get there. So we're gonna just go through a few goals. And if the goal is a skill goal in which you want the means embedded, then you have to just recognize some students aren't gonna be able to achieve that, and you have to embed supports or scaffolds to ensure that all students can achieve the goal. So understanding the goal is a primary starting place. So what we're doing, as I said, this is just raising awareness. And every time a goal is put up there, kind of have that little internal voice saying, are the means embedded? It's okay, but just recognize you're gonna have to support or scaffold it. So I am so sorry, I have, can you, there's, well, maybe we'll just jump here. I had a series of goals that we're going through, and somehow or other this PowerPoints is not the right one that I was working on last night. So we'll go here for a minute. I'm gonna jump to this. This is a goal, which is students will understand the relationship between experimental and theoretical probability. That is typically a middle school kind of goal. Are the means embedded in that goal? No, no means. So that's an open-ended goal in which you can actually say that possibly all students could achieve that goal. If, Kathy, did people pass out that paper that I had for them? People should have a little. Okay, perfect, okay. So we're gonna actually, don't take it out yet. We're not gonna, and paper clips need to be passed out so maybe two per aisle, yes? There is a paper that is a one sheet black and white. Let me show you. Oh my goodness. They're coming, so just ignore that for a bit while we get the tools that we need. So we met with a group of six grade teachers and the discussion started by, what is the problem, math teachers? What is the problem that needs to be solved? So the math teachers told us that their students don't do well on these standardized high-stakes testing in Massachusetts in the area of probability. They don't get it. They don't understand it, they don't get it. So we started out with a problem that they wanted to solve. We knew the goal, but the students were not achieving the goal. So after we had that little discussion is like, what is the problem that needs to be solved? Then we had to go down the path of, what are you currently doing in your math class? So, this is the question, central question. And these are the materials that they were just using. They were using a page from the math workbook and actually that is the printed file that you're getting is the page from the math workbook. That's only, now this is one lesson. Probability of course is a series of lessons over several days, but we chose one lesson to highlight. And they use pretty traditional paper pencil paperclips and they lecture is their method. So it's fairly traditional and how they teach probability. And in fact, it is traditional that they do something called the spinner. I don't know if you have ever done that in your class environment. We're gonna actually try that. So I'll wait until everybody has their little paper. Now just envision, this is in a workbook and it's a good series, but it's in a workbook. And we copied it through fair law, fair use laws. Okay, okay. Is everybody set with their materials? Anybody need anything? Okay, okay. Okay, actually sharing is good because you're gonna share anyway. Okay, so not a problem. Okay, remember, this is the goal and you're gonna put yourself in sixth grade math class right now. Anybody anxious about that? Raise your hand if there's any anxiety about sixth grade math class. Just a few of your, okay, pretty good. Okay, thank you. You do know that math is the only subject in which they talk about math anxiety. They don't talk about social studies anxiety, but math anxiety, okay, that's pretty good. Okay, this is the goal and you may choose, okay, sorry, getting materials all sorted out here. Just like middle school. Okay, it's the students in the back we're waiting for. Okay, okay. You may choose to work alone or with a group of two to four. That's your choice, okay? Is that a mistake in middle school? I don't think so, I don't know. Okay, okay. Okay, so this is your challenge. You have a worksheet, excuse me, I'm gonna just borrow this for a minute Evelyn. You have a worksheet at your disposal and you are to read the math problem. And as you can see on the screen, it's very high tech, it's a pencil and a paper clip. So you should have the tools of the trade and you just put your pencil in the middle, put your paper clip at the point of the pencil and then spin away. And remember where we wanna get to is we want to understand the relationship. So I can ask you, can you please tell me the relationship between experimental and theoretical probability? So work with your partners, we're gonna give you probably about three minutes to do this. Oh, you need a pencil. Oh, I'll sign the one. Okay. It is like middle school. Yeah, it is like middle school. I'm trying to interpret your smile. Oh no, you're very fun with this. Oh, good. All right, that's good. 46. Okay. All right students, are you ready? Remember the goal. The goal is to understand experimental and theoretical in relationship to theoretical probability. Would someone like to be able to be able to tell me about that and what you learned from that task? I noticed a lot of people that were very focused. I also noticed a few of you, I hate to say, were probably doing something else during this time period. And I didn't want to embarrass you by coming over, but I was aware of it and I got my grade book. So we're all set. In terms of the activity, because this and there are some that can't even stop, they're still doing it. And they're ignoring me completely. Okay, in terms of the activity, this actually is an activity that is supported. And in the book that the students use, it is color-coded orange and white. So you have the words red, blue, green and yellow and you're using the colors orange and white. I have a fabulous assistant when she printed this out for me one time. She printed it out with the correct colors. And I said, no, no, Leslie, I want it in orange and white because I actually had to go back and reprint. Okay, the question we want to ask is, what are the barriers to achieving the goal? What did you encounter? What were some barriers? Did it, I mean, this is the goal. What were the, yes? Going yellow, yellow, yellow, yellow. I didn't realize you had the kind of attack. It took you a bit to figure that out. Okay. Sorry, I'm sorry. I shouldn't smile at that. That's great. She lost, and remember, middle school students. At least she only lost the paper clip and didn't flip it across the room. And then also environmental factors influenced the spinning. Okay, other barriers, yes. Wasn't good for collaboration. If you have a classroom, now members are going back up in the States anyway as students. There are 30 students in our classrooms so they would probably have a similar difficulty with closed space. Right. Exactly, okay, good. So it wasn't conducive to collaboration. Other things that were a barrier to you achieving success to get to this. Yes. You needed finger dexterity, absolutely. Because you really, I mean, this is a skill that not all students can do that. Okay, good. Something else, yes. Absolutely, that we know what experimental and theoretical probability mean. Thank you, right, yes. That's right, you couldn't read the instructions. That's exactly right. Because they're hard to read, and not only are they in print, are they exhibiting print disabilities, they're also dense in comprehension disability. So they actually are both small. Okay, other things that you noted. Can we give this enough time? Obviously, I didn't. And in a classroom, you'd obviously have more time. But that's right, good. And you know, for some students, it doesn't matter how much time. Because it's not the time that's the issue, it's the task and the understanding. Yes, exactly. What do you do to plot? Because it, and this is what David was talking about earlier, is construct validity. Is that right? Yeah, thank you, relevance. I knew validity wasn't it. Construct relevance, because what are we really asking students to do? We're asking them to spin. We're asking them to create a table, or a graph, or some visual representation. So we're asking them to figure out the means of collecting data. And actually, one of my colleagues worked with the publishers of this particular activity, this particular series. And what they did is they made it more UDL in that they provided students with a couple of templates for collecting data. So they kept their eye on the goal and didn't let other things interfere. Anything else that might have been a barrier? So this is part of the process, is when you have a lesson that you can almost predict is not going to be successful. You pause, you think about what the goals are, and if it's a goal that embeds means, that's fine. If it's a goal that doesn't embed means, that's fine. You just have to figure out how to scaffold that. And then you have to think about what are the potential barriers? So I wanna show you an alternative way to achieve the goal. And I'll just tell you a short story. Is a colleague and I were in a sixth grade classroom doing a professional development activity with the teachers, but we were just observing. And the students were working on probability. And we said, this is fabulous. They're gonna tell us about probability. So we said to them, what is probability? And there was a small class and they kept saying, and they kept giving us the formula of probability. They said, whatever the formula is, those kids knew the formula, but they did not know what probability was because they hadn't stepped back to figure out, but why? Why are they even learning it? We're talking about authentic learning. Why are they learning it? So I'm gonna show you an activity that we serendipitously said, let's do this with the kids and see if they better understand probability. So let's see if we can all better understand probability. Go here, go here. Do people know the show door site SHODOR? This is a fabulous takeaway if you have people that are in the math arena and they teach students from kindergarten through grade 12, show door off a free site. And it's just another way to represent the information. So I'm gonna go to the show door site. Here we go. This is project interactivate. Sorry, let me just resize this. So this will, what does that look like? So remember the goal. The goal is to understand experimental probability in relationship to theoretical probability. So what we're seeing right here, and I will tell you right up front, there are clearly barriers to this site. I mean there's no question, it cannot read aloud. I mean so we can look at it from a barrier point of view and know that still this isn't gonna work for everyone. But it gets us closer to achieving the goal. So we have the visual representation which is actually fairly true to the colors here. We have a representation of experimental probability and theoretical. Now we can have an opportunity as they say the teachable moment. You can talk to your students about what is theoretical probability? And you can talk about it being a constant in this particular instance. I mean we can create something that is variable here. And now we can start looking at what is it? So let's just do one spin. And what do you predict is gonna happen? Are we gonna get closer to experimental and theoretical probability with one spin? No, okay I'm gonna go, no. Okay so all I do is I have one and I'm gonna say spin, okay? So now what we see is again another opportunity to have the discussion and talk about experimental in relation to theoretical. And we have 100% in the gray, 25%. So let's increase that and now we'll do two spins. And again start thinking and you can engage your students in thinking about the process of understanding probability. So I'm just gonna go with two spins and I'm gonna see what happens. And you can start having them predict what, oh that's right, I did two spins, sorry. And it has a cumulative number so we can see we have three spins. And we spun around. Now students do not worry about the fact you don't see it spin around. They live in a video game world. A couple of teachers have said I didn't see it spin around. But kids don't worry about that. Another opportunity to have a discussion about experimental and theoretical probability. Again we're getting a representation here and I want another representation. So I'm gonna show results frame. So now I get to see and I just wanna minimize my screen a little bit. Okay there, we'll push it there. We'll just have to open and close it. So now you get another representation of what the data looks like. So you can see one third, two thirds. You see it numerically and you see another visual representation. So getting again a sense closer. So how many spins should we go this time? Hundred. Hundred, thank you. Okay so let's, and I only asked you to do 50 before. And remember the goal. See if you can better understand your head should always get wrapped around what the goal is. So I'm gonna just type in 100 spins. And let's, before we jump into it, predict what's gonna happen. What do you think? You said 100 spins, what do you think's gonna happen? It'll, the experiment will get closer and closer to 25 for each. Do you think it'll be 25 each? No, because we have three already and it's experimental, so no. No, okay. I'll tell you a fluky thing that happened once. It worked. It worked on three spins. No, on four spins it went 25, 25. Well that blew the whole lesson. But okay, so let's see how this works this time. So I'm gonna spin. I'm gonna watch this chart because I'll get a visual representation. I'll keep my eye here and let's spin. Oops, sorry. Thank you. Close but really no cigar. Okay, so we're not quite there yet. So who'd like to try something else? A little bit more. Give me another number. Big one. Who's gonna take a risk? Give me a big number. A thousand. A thousand, that's a little bit of a risk. All right, so let's go with a thousand. And then again, remember we're engaging students in discussion and that's exactly what we did in this classroom and they started focusing on oh, experimental and theoretical. When is it gonna happen? So I'll spin here. We have 1103, we're not there yet and I'll give you a little hint. You can spin up to 100,000 times each time. So I'm gonna spin 100,000 and we're gonna see what's gonna happen. Is that 101 more zero? Nope, that's a million, sorry. Yeah, that's it. Okay, almost but not quite. And we were asking you to do 50 spins and see if you could answer the question. And now when we go back to the question, what is the relationship between experimental and theoretical probability? What is it? Talk to me about it. Somebody? What do you think? As the number goes up, the chance of the two meeting. And we said to the students, do you play your parents or anybody play the lottery? What are the chances of winning? All of a sudden it became authentic and they said, oh, we have very little chance of winning. Right, you have very little chance of winning. And they actually bet that you will get more and more because you have very little chance. It became an authentic lesson and I will say in the same amount of time it took us, the students now could understand the relationship. They didn't know the formulas. I mean, and that is what you back it up with that but they understood. Now, this did not work. We had a training once with a fellow who is blind and this was a major barrier, major barrier. He, and we even got the tactile spinner and he finally said, Grace, don't worry about it. I will just be a cooperative partner because that wasn't working at all. That was real. I mean, so that had a major barrier but for more students and that's what you go for. Not every lesson is gonna hit every student. You're increasing the chances of understanding though by thinking about multiple resources.