What I'd like to talk about now is productive struggle and mathematics and computational science, and computational thinking. And I'll share some examples from my own experience as a math teacher and math learner, and also some of the things that I've seen as I've been working with colleagues in an elementary school on computer science. So first the aspects of productive struggle, I see it as a match between the stairs and the crutches, and there are challenges that come from those metaphors. And then I'll share some examples and how I'm seeing it be made visible in math and computational thinking. So the stairs metaphor are those scaffolds that we provide for students, the lectures, the worksheets, the activities that we give them to help them see what a problem is or bridge their thinking to a final solution. But sometimes that support can be too much, and how much is too much, or how much is not enough? If we give crutches, that is, we do the thinking for students, that can lead to frustration and anxiety, even panic if students feel overwhelmed by the mathematics. So some of the challenges that we find in creating environments with productive struggle are to find first of all tasks that are worthy. And that doesn't mean that they have to be tasks that are necessarily applied they can be just math task or just computing task, but things that really require thought. And they have to emerge in a culture of collaboration, that is, where students are free to interact with each other, and able to ask the teacher for the right kind of help. And most important, we need to make explicit the procedures for finding the right supports at the right time, especially for young learners. And I have an example of that and the example comes from box folding. If you take a piece of paper, multiple piece of paper, and cut out the corners for example, if I have a piece of paper like this. That's a grid paper, I can cut out corners that are squares. So, I cut out large squares like this, or, Smaller squares like this from the corners. And then I can fold that up, and I make a tray here. Or with a, With the larger squares that I've cut out, it's a box with higher sides, and the question is, which is bigger, that is, which can hold more? This flat tray, or this taller box with the sides and we'll assume it's just up to the, the volume we're talking about is just up to the top of each, and it's a problem that can be introduced to students as young as elementary school. And it can also be introduced to adults, and they can puzzle over it for a while, which one is bigger? And they can explore it both with manipulatives like this, and also by building different sizes and measuring them. A lot of times they'll use the volume formula length times width times height. And calculate it, many times they will use different technologies like filling it up with rice or with syrofoam. Sometimes they'll make a table and put length times width times height and make calculations and see that there are different volumes and that those volumes do get larger and then get smaller. Can they make a graph out of them, can they use an applet that's on the web? In fact these digital tools adds a whole another level to exploration. So once you make a table in Excel, you can graph the points and see the curve. A Java Applet will let you make different sized boxes and see the volume. A graphing calculator will let you put a curve on top of the points that you plot. And of course, you could use a tool like Mathematica to develop the formulas. The digital tools extend what you can do with manipulatives, and allow you to explore deeper, and give you alternatives. If you're stuck with one approach, you can try another with another tool. So each of these representations, a physical representation, a technological representation, a context of building boxes or the symbols that go with algebra are different representations for the same phenomena. And you can use these to get at your problem and they can help you continue to struggle. Another example which I'll take from my own learning is the famous locker problem, I say it's famous even though it was new to me when I was told about it just a few months ago. And it is one with which I struggled for a number of days. And the problem goes like this, there is a long row of school lockers that have been left open. And students line up, and the first student closes them all, but the mischievous second student opens every even numbered locker. Then the third student goes down and takes every third locker, and opens it if it's closed or closes it if it's open. The student after that changes the state, open or closed, of every fourth locker. The next changes every fifth locker and so on, so then the question is, after say, 30 students, what do the lockers look like? And then what does it look like after 1,000 students? So this is a problem that was given to me by some colleagues who are math teachers, which I puzzled over for a couple of days, and my first inclination was to look at the high tech solution. Or to look at a solution that I could see in technology which was to put it in a spreadsheet and color code the different cells based on how the lockers worked. So on the top row there you see the numbers of the lockers and the columns each row represents a student who goes and changes the state so that number one student shuts everything. And the number two student opens every even one, and the number three student changes every third one, and so on, on down. So you can see the patterns, with the yellow being the open lockers and the blue being the closed. And you notice, or I was able to notice, that something was happening on certain numbers, so I put those in red. Those numbers were 4 and 9 and 16 and 25, and then 36 and then it looked like that same thing was going to happen on 49. So what was going on there? I wasn't sure, these are perfect squares, but why perfect squares? Puzzled over that for a long time and communicating with colleagues, we went back and forth. I'm not sure how much we understood of what each of us was saying. But in the end what helped me was a representation on a good old white board. Looking at the factors I was able to see something is happening with the perfect squares, when we look at all the factors of the numbers. So if we take a number like 5, it's factors are 1 and 5, they pair up nicely, or 8, its factors are 1 and 8, and 2 and and 4, they pair up nicely. But a number like 9, its factors are 1 and 9, and then 3 is used twice, and that's the key, so there is a change of state that leaves it in its same position for the perfect squares. Again, with 15, 1, 15, 3, 5, but with 16, 1, 16, 2, 8, 4, 4 is used only once, and so, the 16, the perfect square remains in that state. And that's how I came to understand that problem and why the perfect squares are the ones that are left finally closed when all the others are open. So those are a couple examples of persistent a productive struggle from my own mathematics experience. And I want to talk a little bit about the culture that is helpful for continuing to struggle. One is one aspect of it is asking the right questions. Students should be able to rephrase those questions, should be able to explain them in their own terms. Their thinking should be logical and they should be able to justify the thinking that they have, in ways that they can explain to others. We can use manipulatives, both technological, that is, digital technology, as well as concrete manipulatives. You should be able to connect the ideas, for example, how you can use what's in a spreadsheet to what's an a table in a whiteboard, and you should be able to help others who are struggling with it. In the elementary school that we were working with, they've made this explicit by giving students what they call the collaborative discussion framework. So when students are having difficulty with coding, another student will ask them, what are they trying to do, what have they tried already? Can they think of something else that they could do? And if the student remains stuck, they could ask a what if question, what would happen if you tried this? Just as a teacher might do it, and the most important thing, which they include at the end, is to celebrate and share once they're successful. And we've seen this happen many times with students, especially with this programming. So here I have a video of a student it's at 15, it's speed up to be 15 times its normal speed. So this was about fifteen minutes of a student's work, what he does is draw a car and then I take the mouse and draw a track, an L-shaped track. And then I ask him, giving him back the mouse, to make the car go from the beginning of the track to the end. And you can see him, he's a fourth grader, programing a joystick to make the car move. So he assumes right away it’s a programing task, and he pulls out tiles to connect the joystick to the car. And as you can see, his initial attempt just makes the car kind of spin around. So he starts again tries a different set of tiles, and perseveres, he's struggling productively. The first time he made it move, but not the way he wanted so now he has to make it move in the right direction. Eventually, he gets the tile in to allow the joystick to move in the direction he wants, and can get the car from the beginning of the track to the end, okay. And we've seen this many times especially with computer programming, that students are willing to persist with problems until they can make it work.