Posted on September 9th, 2011
Margaret (Peg) Smith is a Professor in the Department of Instruction and Learning in the School of Education and a Research Scientist at the Learning Research and Development Center, both at the University of Pittsburgh. In this video she discusses the importance of high-level tasks in developing students capacity to think, reason, and problem solve and the ways in which high-level tasks can provide all students with access to challenging mathematical content.
[Video Transcript] People often ask me how high-level tasks can actually promote equity in the mathematics classroom. And my response to that is twofold. First of all, all tasks are not created equal. They provide students with very different opportunities to think. For example, tasks that ask students to apply memorized procedures to routine problems require one kind of thinking. Tasks that ask students to engage with concepts and to make connections between ideas and representations ask for a very different kind of thinking.
All students need the opportunity to engage in both types of tasks. Applying memorized procedures is important for developing fluency, but engaging in thinking and reasoning at a high level is necessary to develop problem-solving skills. And all students are capable of engaging in those tasks when they're carefully selected by the teacher.
Let's take, for example, two versions of the Odd and Even task. In version one, we see that students are simply asked to fill in the blank after looking at a small number of examples. When they look at the sum of two odd numbers, they need to be able to say that the sum is in fact even. When they look at the product of two odd numbers, they simply need to say that the product is odd. There's nothing else that's required of the student.
When you look at version two, however, a slight modification of the task, we see that students actually have to explain why it does or doesn't work the way it is. They have to come up with a reason why the sum of two odd numbers is even or provide an example that shows that that is not true.
So, students really have to think about what they know and figure out how to frame an argument that would be convincing to somebody else. So in version one, very little is required of students. We would call that a low-level task. It's simply checking on a piece of knowledge that students should have acquired somewhere along the way. In version two, we see a very different kind of task where students have to make sense of the situation and justify the answer that they've come up with. This would be a high-level task.
Now, all students should have the opportunity to engage in high-level tasks and they can as long as the task is at a level that's appropriate for the students and provides them with sufficient access to the problem. So that's a second point.
So let's now take a look at two versions of the Tiling a Patio task. Both versions would be considered high level because there is no explicit pathway given to solving the task, so students do have to figure that out. They have to determine what's the underlying structure of the pattern and how are they gonna use that to explain how you would find the number of tiles in any patio in the sequence. But in version one, students are asked first and only to create an equation that would be used to find the number of tiles in the patio, of any patio.
But, in version two, the task begins by asking students to draw the next two patios in sequence and then to make as many observations as they can about the patios that they have in front of them. Now I’m gonna argue that the first two questions in version two allow students a foothold in the problem. It allows them an opportunity to do something and by trying to get out all the things that they notice about the patios, by attending to how the patios change as you move from one to the next. The teacher then has access to students’ thinking and what they already know and understand about the situation and can ask additional questions based on what they’ve learned.
Now, while you might look at version two and say you’ve just broken the problem down and you've taken away all the thinking, I would argue that you haven't done that at all. In fact, there's still nothing in the problem that tells students what to do. It simply steps them through a process that will engage them in thinking about different aspects of the task, first focusing their attention on how to build the next one, then their attention on looking across the five they have and trying to see what they notice about them. And those two exercises are still all about thinking and trying to reason your way through the situation. From version one to version two of the Tiling a Patio, you see that you have built in some scaffolding for students to actually allow them to make some progress through the problem by working on their own or with their peers in a group setting without having to rely on the teacher to do all of the scaffolding through the problem.
If the same group of students are given version one of the task, there's a really good chance that some of them are not gonna know how to come up with the equation and they're either gonna stop working and shut down or they're gonna start waving their hand wildly in order to get the teacher's assistance so they can get more help on how to work through the problem. And when that happens, the teacher's now in a situation that you have to run around the room and respond to all of the waving hands. And while students are waiting for help, they're not doing anything productive. So I've come to think that the most important decision that a teacher makes in planning a lesson is deciding on the mathematical task that she's gonna put in front of students because it’s the task that puts parameters on what students actually have the opportunity to learn. So the teacher needs to consider both a task that requires thinking and reasoning but at the same time, a task that allows entry to students who bring very different knowledge and skills to the enterprise.
Providing all students with the opportunity to engage in high-level tasks is critical. In fact, in order for students to meet the new expectations as outlined in the Common Core Standards, they must have the opportunity to engage in challenging mathematical activity. In particular, the mathematical practices specify that students need to be able to create arguments, to justify their solutions and to persevere through challenging problems. These competencies can only be developed through work on high-level mathematics tasks.
Margaret (Peg) Smith is a Professor in the Department of Instruction and Learning in the School of Education and a Research Scientist at the Learning Research and Development Center, both at the University of Pittsburgh. She works with preservice elementary, middle, and high school mathematics teachers at the University of Pittsburgh, with doctoral students in mathematics education who are interested in becoming teacher educators, and with practicing middle and high school mathematics teachers and coaches locally and nationally through several funded projects. Over the past decade she has been developing research-based materials for use in the professional development of mathematics teachers and studying what teachers learn from the professional development in which they engage. She has co-authored several books and articles related to mathematical tasks.
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