In the last module, I asked you a couple of questions that I have found in my own experience, engage my own students in thinking about and reflecting on their own ideas. The first question that I ask, involved what's often called the Monty Hall problem or the three doors problem. If you recall, the question was Monty Hall shows you three doors. Let's say, that you choose door number 1, and Monty Hall opens door number 3, showing that there's a booby prize or a goat behind that door and then ask you, "Would you like to stay with your initial choice of door number 1, or would you rather switch to door number 2?" So then the question becomes, would you have a higher probability of winning if you stayed with door number 1 or if you switch to door number 2? Most students, or when initially encountering this problem say, "Well, it's obviously 50-50, there are two doors so it's a coin toss." We then moved on to looking at this computer program, that showed that in fact it is better to switch. You would win about one third of the time if you stayed, and when about two thirds of the time if you switched. So I'd like to ask you, hopefully you've had a chance to think about this, discuss with your classmates, perhaps look online on some explanatory ideas. Were you able to make sense of the result that it is better to switch and if so, what was helpful? So please take a few moments to type in your thoughts into the text box. So one thing that I typically do after showing the results of the computer program because students often are not convinced by the computer program or they're perhaps convinced that it's better to switch but don't really understand why, it doesn't make sense to them, has involve them in a rather extended period of discussion about this idea. Students are sharing ideas and some students are arguing that, it's still better to stay. It's better to or it doesn't matter. Other students are beginning to get an inkling of maybe it is better to switch. One example of something that students will often try to do, is something like this table here. Well, they'll talk about all of the different possibilities. So if you look at this table, sort of discusses all of the possibilities. Your initial choice is door number 1, 2, or 3. If you choose door number 1, the prize could be behind door number 1, 2, or 3, etc. So enumerating all of these possibilities, we can see that there are three instances where you would win if you stayed, but six instances where you would win if you switched. Students will perhaps do a table like this or show some kind of a tree diagram. What I've found is that, when this happens, other students who are listening to this presentation, again, might be convinced as with the computer program that it's better to switch, but it still doesn't make sense. It's almost like empirical arm twisting and logical arm twisting, but it still doesn't make sense to them. Often then we'll go on to another question, which does seem to help a lot of students make more intuitive sense of this context, and that is the following. Let's say, rather than having three doors, let's say, that you had a 1,000 doors. Let's say, that you choose door number 210. Now Monty Hall's rule if you will, is that he's going to open up every door except for the door that you initially chose and one other door. Now this is the rule that he followed in the three door problem, is just needs to be more explicitly expressed in this way in the 1,000 door problem. So he knows which door the sports cars behind. So he's not going to open that door. But he's going to open up every other door. So let's say, that he opens up every door, except for your initial choice of 210 and door number 643. Now, do you think that we'd be better to stay with your initial choice or to switch? This begins to get students thinking that, "Well, it's really unlikely that the sports car would be behind my initial choice, like 1 in 1,000. In fact, it would be a 1 in 1,000 probability that my initial choice would be the right one." So maybe, it would be better to switch. Even though, I mean it is still seems sort of like it's 50-50, because there's two doors, but then again, it's really unlikely that my initial choice was the right one. So maybe, it would be better to switch. So these two intuitions begin to battle with each other. Through discussion, the intuition that my initial choices unlikely, in this case, highly unlikely, 1 in 1,000 as opposed to just 1 out of 3. Students can begin to make sense of this, not because they've been empirically are logically arm twisted, but because they're drawing on another intuition that can help them to make more intuitive sense of the idea. The other question that we looked at, was this question of what we'll see on the bottom paper. So if you're sitting in this chair in this classroom and you have this piece of paper, and let's see what we observe when we raise up this piece of paper. So most students predict that you'll see one spot of light coming through the hole. But we in fact see a number of spots. So this is a very puzzling result, it's very surprising to most students. So here are the observations that we see asking what do we see on the bottom paper. We notice first of all, there are multiple spots as opposed to just a single spot, which is what most students predict. We see that as the top paper moves up, the spot spread out. We see that if the back lights are turned off, the front spots disappear. This isn't something that we tried here, but it is something that does happen if we try that. Further the hole is round, but the spots appear rectangular. So let me ask this, were you able to make sense of these observations? If so, what was helpful? So again, take a couple of minutes to write down your thoughts in the provided text box. So I'd like to talk now about some of the interactive discussions that I've had with my students when I introduce this question, which again is surprising. Students do take some time to try to make sense of the ideas. So what I'm going to show here are some screen captures of an iPad that I've used, that projects onto a screen that everyone can see. So students can pass the iPad around and use the iPad to express visually some of their thoughts. So here's a first step at trying to come to an understanding of this these observations, which is more of a visual representation of the observations rather than an attempt to explain them. But it's a start. So you've got lights on the ceiling and light goes through the hole and it creates these multiple spots. Now here's a student try to come up with more of an explanation. Her explanatory idea was that light is coming through the hole then because of little pieces of paper around the hole that was made that perhaps those are splitting the light and making them form these multiple spots. So that seemed like a really good idea to test. So we tried it, we took a flashlight and shown it through the hole to see if it would split up into multiple spots and lo and behold, there was just one spot when we shine a flashlight through the hall. So students seem to think, "Well, that makes it difficult to accept that model." Then another student thought, "Well, maybe it has something to do with the multiple lights on the ceiling and maybe the lights are coming from each of those lights, and they're coming through the hole and then bouncing around, and eventually making a spot." Another student modified this model to the idea that, there are multiple lights on the ceiling, but the light coming through the hole isn't bouncing around, is just going directly to the piece of paper underneath. We talked quite a bit about this model and began to get more and more excited about it. So let's take a look at this model and whether it explains some of our observations. So the first observation, is that there are multiple spots. Let's see if this model helps us to explain that. So yes, it looks like on the bottom paper there are multiple spots, and each one formed by light coming from one of the lights on the ceiling. So we can check that first observation off. The model seems to explain that. As the top paper moves up, spots spread out. Let's look at this. Now, this is drawn not with the top paper moving up and down but rather with the bottom paper moving down and up. But it's still the distance is getting more and less. So we can see what happens when the distance is close to spots are close, as you can see at the top line there. As the bottom paper moves down, or the spots do spread out and they spread out quite a bit as the paper goes way down. So this model does help to explain why the spots would spread out as the top paper is moved up and down. If the back lights are turned off, the front spots disappear. Will see if this model explains that. So there's an arrow pointing at one of the top lights there, and when that light goes off, you can see that the corresponding spot, this model would predict that the spot on the left would go off. So if the top light was the back light, and the front spot would go off and vice versa. So this model does help to explain that observation as well. What about the hole is round, but the spots are rectangular? Maybe not so much. Each of the lights is represented as a circle and it makes a circular spot. So it helps to explain the first three observations, but perhaps not so much that observation. So let's focus now on that observation that the hole is round, but the spots are rectangular, as you can see in this screen capture. Now, here's a student attempting to make sense of that and drew a rectangular light, but showed that, if this way of representing it seems like the light coming through should still just be round. It should still take the shape of the hole that it's coming through. So this raised the question of, how can we make sense of this? Here's another student who pointed out that, when a flashlight, an LED flashlight in this case, or a flashlight, an LED light on a phone, when that shines through the round hole, the spot is round. But when a rectangular fluorescent light shines through the hole, the spot is rectangular. So this is again, pointing out those observations without really providing much of an explanatory model. But it definitely, it frames the question in a nice way. Here's another student trying to make sense of this by drawing a line to represent a rectangular light, and a spot to represent the flashlight. So at this point, I thought these ideas are being expressed in batted around. Let me as the teacher, maybe take a shot at formulating the question, not answering the question, but formulating a question in a way that perhaps might help us to make some progress. So I asked the following, I only drew the top two sets of lines, and a spot on the left and a line on the right, to ask the question when we have a single flashlight, why is the spot round, and when we have a rectangular fluorescent light, why is the spot rectangular? Right after drawing this, one of the students, "Oh, I know," and she was very excited, and she drew what we see here, showing that light coming from the single spot of light, the LED flashlight, would just go through and make a round hole. But for the extended light, light rays from each part of that light would be shining in all different directions, not just as we see here, not just the left-hand part of the light, the light would be shining to left, and the right-hand part, the light would be shining to the right. But rather that at each point on the light, light rays would be coming from that and shining in all different directions. Which violates the drawings that we see in most textbooks, where light rays are shown. As we see here, coming out of light and just spreading out. So she was saying, okay, by that way of thinking, the light rays coming from all the different parts of the light, would act like, if the rectangular light were an array of lots of spots of light, and it would form lots of spots of light on the bottom sheet, and therefore, create a rectangular array on the bottom sheet. The drawing on the bottom shows she's articulating that, this was hard for her to think about because, she had to break away from this way of thinking about the light and think about light from each of the spots as going in all different directions. So it was a real feat of coming up with a model, and critiquing that model, and arriving at something that helped to explain this final observation that the hole is round but the spots are rectangular. So some implications of these reflections on discussions of questions that you've thought about and that we've now thought about together, that even in contexts that are completely new, such as the Monty Hall problem, the first time that you heard it, or the hole in the paper which is probably new to you as well. Students will often have ways of making sense of these contexts and have strong intuitions about what should happen. Some student ideas will conflict with instructional goals, and some will support these goals. As we saw with the Monte Hall problem that, the intuition of 50-50 argued against the result, whereas, the intuition that it's unlikely that I'll pick the right door to begin with, helps to support that. Those ideas can go back and forth, and help students to make intuitive sense of the difficult ideas. Further that through interactive explorations and discussions, students can draw out in support grounded ways of sensemaking in these contexts. Further, telling that does not tap into ground in sensemaking is typically ineffective. As we saw with the Monte Hall problem, where very careful articulation of an enumeration of all of the possibilities was really not that helpful for most students. Teachers can help focus sense making activities, as I did when I reframed the question from the separate contexts that they were, the students were constructing and put it into one context and asked, "What about the flashlight on the left and the rectangular light on the right"? Then, that's helps students to see the problem in a new way and come up with new ideas. So we've had a chance to reflect on some of our own sensemaking as learners, and so next, I'd like to look at examples of learning environments that support sensemaking, that others have talked about.