Welcome to the second module on constructivism. Last time we talked about some common sense versus constructivist assumptions, and the first common sense assumption is that I should focus on my actions as a teacher in providing good instruction, and the second one that I can get my ideas across to my students if I'm clear. The constructivist reframing of those assumptions were my focus should be on my students and their ideas, and I can't get ideas across, but I can help my students engage and modify their ideas. So it's clear that from a constructivist perspective, there's a huge focus on students ideas and idea based interactions. So in this module, I want to explore with you this idea of ideas. So to begin with that, I want to ask the question, what is an idea? To begin to address that question, I wanted to start off by asking you a question that hopefully will engage your ideas so that you can get a sense of what is an idea from the inside. So let me just set the stage. This is a picture from a game show from a number of years ago in the '60s or '70s, and in the center there is Monty Hall with a microphone. This is a game show where contestants dressed up in wild costumes and did crazy things. But one of the things that Monty Hall often did was he would ask someone to look at three doors or three curtains on the stage, and ask them to choose one of those doors. Now, what's assumed Monty Hall has said to the contestant that behind one of these doors is a fancy sports car, something that you would really want. Behind the other two of these doors is a booby prize let's say a goat, and now some people would prefer a goat to a sports car, but let's assume for now that you would prefer to get a sports car, and so a goat would be something you wouldn't want. So let's say that you choose door number one, and then Monty Hall in order to make things more interesting. Now Monty Hall knows which door the sports car is behind, which doors the goats are behind. So he's going to open up one of the doors that has a goat behind it. So he opens up door number 3 and says, "Okay, I'm showing you door number 3, and that has a goat behind it. Would you like to stay with your original choice of door number 1 or would you like to switch?" So I would like to ask you a question, which do you think would benefit you more? would it be better to stay with your original choice of door number 1? Would it be better to switch to door number 2? Or would it not matter? So I'd like to ask you to take a moment to write down your thoughts in the provided text box, and we'll continue when you finish that. This is a problem that is it's actually rather famous. You may have heard of it before or you may not. If you have heard of it before, then try to think back to when you first heard of it. If you haven't heard of it, then just pay attention to how you're thinking, how your ideas develop, etc. Now, this is a question that when I ask it of my students, students who haven't heard about the problem before almost invariably say, "Well, it shouldn't matter." One response that I often get is, "Well, I think I'd rather stay, because if I switched and I lost I'd feel really bad." So we reframe that to think about probability in which case when you have a higher probability of winning if you stayed or if you switched. When framed in that way, most students say, "Oh. Of course it would be 50/50, there's two doors. So it doesn't really matter, it's flip of the coin." So we go on to say well, how could we explore this more? So we try a simulation, which when I'm doing it face-to-face, we have our students pretend to be Monty Hall, and they'll go around with three cards and lay them on the table and each person in the class becomes a contestant, and the person will choose a door. The person playing Monty Hall will turn over one of the cards that has a goat on it, and ask if they would like to stay or switch. The person will then decide to stay or switch, and then we'll see whether they won or lost. So since we're not face to face, I'm going to run a computer simulation that simulates that kind of interaction. So I click on run and type in the number of times. Let's try to run it five times. Let's say that our students playing Monty Hall is doing this with the first other students in the class. The contestant chooses door number 1. Monty Hall says, "Okay, I'm going to open door number 2 and show you that there's a goat behind it." The car is actually behind door number 3. The contestant switches to door number 3, and of course wins because that's where the car is. Moving on to the second student, the contestant chooses door number 1. Monty Hall opens door number 3, and the car is behind door number 1. The student decides to switch to number 2, and since the car is behind one, they lose. So you can look at the other ones in there, they're all very similar. So just a summary at the bottom, the contestant switched in one twice, the contestant switched and lost twice, stayed in one zero times, stayed and lost one time. So it was better to have stayed two out of five times. Now, students typically say, "No matter how many students are in the class this doesn't really give us anything definitive because there are so few." So we decide that well, if we had a computer program that could do it 1,000 times or 10,000 times, then maybe that would tell us more information. Students would say, "I happened to have that." So with this simulation, when we run it, it will not show all of the individual times. It will just show the summary so that we can run it as many times as you want, and just get the summary. So running it a 1,000 times, we can see that it's better to have stayed about 314 out of a 1,000 times for that particular run. Let's run it 10,000 times, and we can see that in that case it was better to have stayed 3,346 out of 10,000 times. Running it a million times, we can see that it was better to have stayed 333,538 out of a million times. So in each of these runs, it was better to have stayed in approximately one-third of the cases, which means that it would have been better to switch two thirds of the time. So the answer to the question according to this this simulation is that it's better to switch. It gives you a higher probability of winning. So when I do this with my students in a face-to-face class, I get to common reactions. One reaction is, okay I guess it's better to switch, I guess I'll accept that, but it doesn't really make sense. Another reaction that I often get is, well, no, that's not right. There must be something wrong with the computer simulation. It has to be 50/50 because there's only two choices. So rarely do I get the response, it's better to switch. That makes sense. That kind of empirical results just doesn't seem to help students to make sense of the idea. But I would encourage you to explore online and discuss with your classmates. If you google Monty Hall Problem or three doors problem, you'll find literally 100s of discussions of this particular problem that you can look at, and go on the discussion board and discuss your ideas with your classmates. We'll look at this more in the next module, where I'll talk about the kinds of discussions that we have about these ideas, and how we go on to try to make sense of the ideas. Another idea that I often engage my students with is in the following context, and this is a picture of a classroom that I often teach in. The question that I'm going to ask is of a student sitting in this chair. The question that I ask is the following; let's say that I have a piece of paper and I poked a hole in it with a pencil so it's a small round hole in the piece of paper. Then I have another piece of paper underneath that. What I want to ask is, what do you think will happen if I raise the top piece of paper with a hole in it off of the bottom piece of paper? What will I see on the bottom piece of paper? So this is a drawing of that, and I'm not asking what will I see if I look through the whole, what I'm asking is what will I see here on the bottom piece of paper as a raise the top piece of paper. So I'd like to ask you to take a minute to make a prediction with the reason, what you think you would see on the bottom piece of paper. So again, what do you think you would see on the bottom piece of paper? A very common prediction, in fact probably 99 percent of students that I asked this of make the following prediction, that I would see a spot on the bottom piece of paper. There's a hole. So some light should come through the hole make a spot on the bottom piece of paper. Perhaps, they might go on to say if I raise the bottom piece of paper, the spot would get larger, and maybe a couple of other things. But what do we observe? Let's look at this following video and see what in fact happens. So what are some observations? I think one of the most obvious observations is that there are multiple spots, it's not a single spot. Another observation is that as the top paper moves up the spot spread out. Another thing that we often try when I do this is I turn off the back set of lights, and what students notice is that the front spots disappear. Another observation is that the whole is round but the spots are rectangular, as you can see in this frame grabbed from the video. So how can you make sense of this? So here you are sitting in this room. How can you make sense of these observations? So I would like to ask you to again explore online, discuss with your classmates. There's not really an easy thing to google here for this particular problem, but you might be able to think of some things, but discuss with your classmates. See if there's a way that you can make sense of some or all of these observations. So just to recap, again we've had a chance in this lecture to think about what is an idea by exploring our own ideas. I think a couple of things that we have seen is that ideas often involves strong expectations. Now, I know when I do these two activities with my own students, a very common response is, why are we even doing this? Of course, it's 50/50 or of course it's just a single spot, that that goes without saying. Then they're very surprised to see that those expectations are not met. Ideas often involved cultural aspects. So ideas of 50/50 or a coin toss, or those sorts of things are obviously come out of a culture that thinks about gambling and probability, and that sort of thing. Intuitive ideas could also be called conceptual attractors. I'll be talking about this in our future lecture. Then our thinking seems to be attracted to particular ways of making sense of situations. Also ideas sometimes support and sometimes oppose development of more sophisticated expert ideas. Now, we've been looking at some examples of ideas that might oppose the development of sophisticated expert ideas, but there are also other ideas that students have that can provide intuitive support for expert ideas. Another thing that we've seen, particularly in the first module is that teachers are often unaware of their students ideas. If you can recall the teacher in one of the first lecture is saying, "This is mind-boggling," referring to one of her students ideas. We'll be seeing some more examples of ideas that are somewhat surprising. So in the next lecture, we'll be looking at some examples of student ideas.