Going deeper: How do open tasks encourage a focus on learning?

Open tasks focus students on their learning process and deep understanding of the material. In a growth mindset task, students can be asked to find patterns and to gain a conceptual understanding prior to a mathematical understanding. Let's see an example from Professor Jo Boaler.

Professor Jo Boaler, Mathematics Education Expert, Stanford University: So here is a great task that can be very different depending on how it's presented. So first I'm going to present the task in a way that is oriented towards a growth mindset. So, in this task, I'm going to ask you to do, you're going to, you'll see Case 1, 2, and 3 of a shape growing, and my question for you that I want you to think about and write about is how do you see this shape growing? How would you describe the growth?

So when we use this task when we were teaching algebra to some seventh and eighth graders, a group of three students saw it grow in two different ways, as you can see from a picture of their work. With one of the boys saw it growing as cubes adding on to the bottom row each time, and the other one saw it as cubes adding on to each column. Now there are other ways of seeing the growth that you might have seen. Some people see it as a growing square, for example.

But as students worked on that task, and you also asked them to work on how many cubes were in the hundredth case, they really had to think about the shape and the form and the number of cubes, and as they worked on that, they came to develop and understand a quadratic function. So that's really big. They weren't just given a function to manipulate, but they are actually forming it. They are seeing it visually and understanding its components.

And as the students worked on their task, the first thing they realized was that they'd seen it in different ways, and they talked about that, and that deepened their understanding of the function. They then worked to try and work out how the shape was growing.

When I've worked in workshops and I've given this task to teachers with the same wording, asking them to find the hundredth case, thinking they would work in similar ways to the students Instead they made a table, put in the numbers, could see that the function was growing as an n + 1 squared funciton, put in the number for the hundredth case, and came up with an answer.

Then when I asked them what their n represented and their n + 1 squared, they really didn't know what the n represented, and I realized that that's changing the presentation of asking people how do you see the shape growing and how many cubes are in case 100. It completely changed the task.