[MUSIC] So now that we've had a chance to kind of look at these constructivist learning environments from the inside as learners, I'd like to look at some examples of constructivist learning environments that others have instigated. And to look at what those look like, what are some of the results. But to start with, I'd like to look at a learning environment that is not constructivist, that is based on behaviorist perspective on learning. And so this is individually prescribed instruction. This was an individual, sort of an individualized approach to mathematics teaching that was based on a behaviorist view of learning. That is, the shaping of behavior through immediate feedback. So similar to how I learned to type, students are taught very explicitly the procedures that they are supposed to do, and then they practice those procedures until they get good at those procedures. And so the mathematical procedures are taught directly, typically through example calculations, here's an example of adding fractions, here's another example, and so they sort of get the pattern that way. And the students worked individually at their own pace on worksheets to practice the procedures. And students took frequent quizzes and got immediate feedback on the correctness of their answers. 80% correct is considered mastery and then they can move on. And they can retake a quiz if they don't achieve 80% correct. Now let's look at Benny, who was the sort of case studies student in this. He was a sixth grade student, and he had used IPI since the second grade, so he was very experienced with this approach. And he was making much better than average progress in terms of getting mastery on quizzes and moving quickly through the program. The teacher considered him as one of her best students in mathematics. The author of this paper, Earl Langer, talked with Benny. And rather than just giving him problems, and asking for solutions, and giving him a feedback on his answers, he asked him, how do you understand this? He probed his thinking, as we saw with interviews, and what he found was rather surprising. Here was Benny who was making very good progress, but he had a lot of alternative mathematical rules. And here's just one example, converting fractions, he would add the denominator and numerator, and place the decimal based on the number of digits in the numbers. So for example, 429 over 100 would be 5.29. You add up the numbers and get 529, and then place the decimal, which would be two decimal places since we're dealing with hundreds. 3 over 1000 would be 1.003, you place three decimals since you're dealing with thousands here, etc. As you can see, he was very consistent, but the rule was obviously misconceived. How could this be? How could he make such progress if he had such incorrect ideas about these rules? And it had to do with his view of mathematics. His view of mathematics is if there's 100 different rules, even though the answer is right, you might need to express it differently to get it right by the key. And the key is what the the teacher or the teacher's assistant would use to grade his quizzes and get him very quick feedback. Example that he gave is 2 over 4 is the same as 1 over 2, but I would get it wrong if I wrote 2 over 4, and so I need to write it 1 over 2. And so even though two answers might be the same, you have to express it differently if you get it wrong by the key. And so different contexts call for different rules, and so here he's talking about that. So for example, if you have 2 plus 3, that's 5. If I did 2 plus 0.3, that will give me a decimal, that will be 0.5. If I did it in pictures, that will give me 2.3. If I did it in fractions, like this, that will give me two and three-tenths. And these are all the same to Benny, they're just different expressions of the same answer. And so this is why he was able to move quickly through this program, even though he had some obvious misconceptions in terms of not just what the answers would be, but what the answers represent. To him, they were meaningless symbols, that you could have this symbol or this symbol, and you just have to know the key says you should have this symbol. He wasn't thinking of these as mathematical quantities that he could reason about and make sense of, but rather as answers that were kind of meaningless to him, but he just had to guess, he called it a wild goose chase at one point. Moving on from this unfortunate example of kind of the failure of an approach based on the common sense that worked well for teaching typing, but obviously does not work very well in a more conceptually rich area such as mathematics. Let's move on to looking at some examples where students' ideas are taken very seriously and the sensemaking process is taken very seriously. So here's one example, a constructivist approach to second grade mathematics. And in this approach, there was an attempt to apply a constructivist perspective to second grade mathematics instruction. And they worked with one class for a whole school year. And the instruction was problem-based, problem solving in pairs, and they needed to reach a consensus on the problem and discussion of problems with the whole class. And there's a distinction here that needs to be made between a problem and an exercise. An exercise, similar to what Benny was doing, is you teach students here's the correct procedure and now here are a number of exercises that you can use to practice that procedure until you're really good at it. A problem is a genuine problem, something that you're presenting to students and they don't know a solution method. They have to figure it out, they have to make sense of it and come up with a solution method themselves. And so this was the focus of this second grade classroom. And so multiplication problems, they weren't shown how to do them, they were given multiplication problems, for example, and they had to figure them out. And so they had to make sense of, not just come up with answers, but they had to make sense of why their answers were the way that they were. The teacher was very authoritative about the classroom culture, but very open about the mathematics. And so the kinds of discussions that were held about what are we doing here were very different than let's think about the mathematics. And we'll see some examples of that. And the structure was one of maintaining a safe environment for sensemaking. The teacher was, again, very authoritative about making sure that the classroom culture was one that valued students' ideas and sensemaking. So let's look an example of the teacher being somewhat authoritative about creating a learning community culture. So one of the students asks, how many problems are there? The teacher says, there are eight. So the most you can do is eight. Some of them are very hard, though, so you may only get two done. Is that okay if you only get two done? Now, in a traditional classroom, the answer would be no, you gotta get all of them done. But here the answer is, the students all realize that yeah, that's fine. The teacher says, you bet it is, if you only get one done because it was so hard and you worked so hard on it that it was the only one you got, that's okay too. And then one student says, if you got none done, but you're still working on it, it's okay too. And so this was a very different culture, that the focus was not on getting all the work done. But the focus was on learning, making sense, understanding, solving problems, thinking hard, that sort of thing. Here's another way that the teacher sort of created the learning community culture. Okay, you're going to cooperate, you're going to work with your partner. And if you figure out the answer and your partner is looking at you like, how in the world did you get that answer? It's going to be up to you and your partner to work it out and understand it together. Then you can get another problem card, but not until then. The culture in this classroom was not that each pair would figure out the answer and then come get another problem. But rather that the pair would work together until they both understood why the answer was the way that it was. Here's another example, where the teacher is very authoritative in trying to quash something that she was seeing that she thought went against the learning community culture that she was trying to establish, that valued students' ideas and didn't make students feel bad about expressing their ideas. Now, another thing I noticed was happening, and it's something that I don't like and I don't want to hear, and it's these two words. These words are no-no starting today. What are these two words, Mark? That's easy. I've had kids come up to me and say, that's easy. Maybe I look at it and say, I don't think that's very easy. How do you think that's going to make me feel? It hurts my feelings when someone says, that's easy. When I'm struggling and trying so hard, it makes me feel kind of dumb or stupid. Because I'm thinking, gosh, if it's so easy, why am I having so much trouble with it? So these are examples where the teacher was very authoritative and was very much involved in a very directive way in creating a culture that valued students' ideas and that encouraged students to express, critique, and modify their ideas. Now, here's a very different example of a sensemaking discussion. And in this discussion, the children were discussing, with the teacher, the meaning of seven-eights. And John had a problem that he was thinking about and he brought this up, but what if it is one and one, what would that be? And the teacher said, like this? And she writes 1 over 1 on the board, good question. Or what if you had this over this, she writes 4 over 4 and 6 over 6, what does that mean? So again, she's sort of putting the onus back on the students to make sense of that. John says, put a circle down. Okay, put a circle down. And then she draws a circle on the overhead, then looks at John. Like, what I'm thinking is one and one, he points to 1 over 1. How could you make like, he looks at the teacher. And the teacher, rather than try to answer his question, puts the question back to the class. How could you show one and one, 1 over 1? John looks at the teacher and says nothing. Then another student interjects, just fill it all in. Yeah, that's one piece. Okay, if we remember what a fraction tells us, John. John interrupts the teacher excitedly, but it's the whole thing, not just one piece. So how can we fill it in? The whole thing! So we can see this is a very different feel to the kinds of authoritative instructions that the teacher had in trying to establish the classroom culture, where now this allowed the space for these kinds of discussions to take place. Some outcomes of this, first of all, the children were enthusiastic and persistent, and they looked very closely with videos. And the students expressed joy when they solved a personally challenging problem. And here's one result that I found almost difficult to believe, but in thinking about my own experience, that kind of makes sense. Now, this was in the second semester, and so they're not saying the first semester there were some issues getting students used to this kind of environment. But in the second semester, they were not able to identify a single instance when a child evidenced frustration while problem solving during that second semester. In fact, the children often requested additional time for problem solving. And so they tended to allocate three consecutive one-hour class periods to general problem solving. And so this might seem almost too good to be true, but it reminds me of an interaction I had with my own children, and this is when they were in elementary school. And I was sitting with them, I have three kids, and so there were four of us. I was sitting with them and they were all saying how much they hated math. And so I was concerned about that. And so I asked them a question, a genuine problem question rather than an exercise. There are four of us here, what if each of us gave each other a hug, how many hugs would that be? And they came alive, they were starting to draw pictures, and hugging each other, and counting hugs, and talking about all different ways of thinking about this problem. And when I said, well, this is mathematics, and they said, no, that's not math, math is solving sums and doing exercises. And so it was really kind of disconcerting, but it was really interesting to see how excited they were when they were faced with a genuine problem, not just an exercise to apply a previously taught procedure. Further outcomes, the achievement scores rose satisfactorily, and by the middle of the school year, parents and administrators were universally supportive. And the program was expanded to 18 teachers from the same school system. And so this is a pretty strong result, if something like this gets high marks from the administrators and parents, then that's a pretty good outcome. Another example is middle school mathematics. And this is paper that came out this year from Joe Boaler, who has looked a lot at elementary and middle school mathematics and the kinds of problem solving environments that we've been talking about. And so this was an 18-day summer program, the mornings were spent on mathematics, and the afternoons were spent on other interactive kinds of things, not dealing with mathematics. 83 middle school children from a variety of backgrounds, but all of the students described themselves as not a math person. In fact, when they chose students for this program, they wanted to make sure they were students who were struggling with mathematics and had views of themselves as not very positive in terms of math learners. And again, there was a focus on problem solving. And so here's an example of a problem that they looked at. And so you have a square piece of paper, and how can you get these different shapes and prove to others that you have that shape? So for example, looking at the first one, construct this triangle with exactly one quarter the area of the original square. Convince your partner that it has one quarter of the area. So similar to what we saw in the previous article, where they needed to come to a consensus, here they also needed to come to a consensus and agree that the problem is solved appropriately. Similarly, if they're looking at something like 1 divided by two-thirds. In a procedural classroom, you would follow the rule, invert and multiply, and then you're done. In this classroom, it became a genuine problem. How can we make sense of why the answer is what it is? And so students would be drawing, they would be discussing with each other, arguing, trying to prove to each other that their way of looking at it is an appropriate way of looking at it, etc. And so here is a discussion with a couple of students at the end of this program. So what did it take in summer math camp to be successful? Being able to communicate with your partner as you go, and being able to show visuals, not just numbers, being able to explain things well. And then someone says, how or why or, and together they say, prove it, and start laughing. And so there was clearly an emphasis on coming to a consensus and agreeing that the answer is what one person is claiming that it is. And so some outcomes of this program. They had a standardized pretest and posttest. And what they found was that the average improvement after 18 days in the summer program was 50% on this particular standardized test, which represented an equivalent to 2.8 years of school. And so here we have students who are, in many ways, at risk of having difficult time in mathematics. And in only 18 days they're able to make incredible progress through these processes of problem solving, and sensemaking, and discussing with each other, expressing, critiquing, and modifying their ideas. And they began to feel much more like they were competent at mathematics. And they began to say, being a good math person isn't just being the first one to have your hand up, but it's being able to think mathematically to reason things out. And so these were very, very exciting outcomes. In the next lecture, I'd like to look at some examples of similar kinds of environments from science. [MUSIC]