Content of Thinking
Use Rigorous Tasks
To prepare future teachers well, it is not enough to expose them to ambitious content. Programs must organize preparation around rigorous instructional tasks and ensure that these tasks are modeled and unpacked by teacher-educators and practiced by candidates.
By “instructional task” we mean the cognitive work that students are actually asked to do in response to content. Rigorous tasks include those that:
- Build on and extend prior knowledge
- Involve effortful thought
- Deepen conceptual understanding of material
- Require learners to communicate their understanding to others
- Involve multiple interpretations or paths to a solution
Too often, we see ambitious content dumbed down through low-cognitive demand tasks. In one elementary classroom that we recently visited, a beginning teacher was using a well-regarded English Language Arts curriculum that asked students to write a five-sentence analytic paragraph in response to a short story. But when it came time for students to write the paragraph, the teacher changed the directions. She asked students simply to write a complete sentence using the language of the prompt. Ambitious content morphed into an exercise in regurgitation.
“Teachers can be really good at some skills or practices,” says Dr. Ellen McIntyre, dean of the Cato College of Education at the University of North Carolina Charlotte. “But if the task is not meaningful and high-quality, it can be a pretty useless waste of time in a classroom.”
Over the past three years, Deans for Impact has worked with Dean McIntyre and her team as they redesigned their teacher-education programs. This redesign has included close attention to the critical relationship between content and instructional tasks.
Dr. Luke Reinke, an assistant professor of mathematics education at UNC Charlotte, knows the vital role that rigorous tasks play in fostering true conceptual understanding of complex mathematical ideas. Professor Reinke’s upper elementary math methods course is organized around a series of candidate-led rehearsals of ambitious, high-leverage content. Each week, a small group of candidates co-plans a lesson with Reinke, receives multiple rounds of feedback on their written plans, and then “rehearses” the lesson with their peers standing in as elementary-age students.
Drawing upon North Carolina’s curriculum guide and the Common Core State Standards for Mathematics, Reinke chooses content that is essential to developing a young learner’s mathematical reasoning abilities.
Importantly, he pre-selects the instructional tasks himself, rather than ask teacher-candidates to find them on their own.
“I want to offload that responsibility at the beginning,” Reinke says, “so they can get a chance to teach tasks that are rigorous before they actually think about, ‘Okay, how am I going to select these for myself?’”
Ask the average American adult to multiply two fractions – say, 1/3 times 1/2 – and you will probably get one of two responses. The first, a befuddled look. The second, what is called the standard algorithm: multiply the numerators across, multiply the denominators across, and you have your answer.
But now ask why this algorithm works, and you’re likely to get a blank stare.
This is because most adults in the United State have learned multiplication of fractions through low-cognitive demand instructional tasks. We were taught the procedure, not the mathematical reasoning behind it. We didn’t learn that multiplication of fractions builds upon multiplication of whole numbers. We didn’t learn that if we use multiplication to calculate three groups of four, then it makes sense to use multiplication to calculate half of a group of four, or one-third of a group of one-sixth.
We were asked to produce the right answer, not to communicate our understanding of the concept. Of course, knowing the algorithm is helpful in building automaticity and lowering cognitive load. But as mathematics gets more complex, failing to have a conceptual understanding of why the algorithm works can make it hard to learn new mathematical ideas.
There are multiple ways to teach multiplication of fractions more rigorously, but one way is to activate prior knowledge of multiplication and then introduce a visual or physical representation, asking learners to show how they might multiply two fractions using their understanding of whole-number multiplication.
This is how Reinke decided to begin his co-planning session with three of his teacher-candidates – Natalia, Tim, and Nicholas: For a longer look at Dr. Reinke's pre-planning session, click here.
For pre-service teachers who have been exposed only to a procedural way of multiplying fractions, conceptual understanding is a massive leap. “When Dr. Reinke first gave me these problems, I was looking at them and just really honestly looking at him to try to find the answer,” says Natalia.
Using rigorous instructional tasks helps pre-service teachers connect concept to procedure, and anticipate how children will understand the content. In the case of multiplying fractions, Reinke and the teacher-candidates adapted an existing curriculum to create a task where their classmates – as elementary-age students – had to analyze an authentic scenario, draw a picture, create an equation, and then communicate how the picture was related to the equation.
“It’s trying to get them to make sense of that standard algorithm in terms of a visual representation or a real life scenario,” says Reinke.
Here’s the real-life scenario that Natalia, Tim, and Nicholas created with Professor Reinke:
Dr. Reinke makes a full pan of brownies to bring to school. He’s going to bring half of the pan to his morning methods section. He knows a bunch of the students like icing so he’s trying to figure out how much icing to make. If he ices one third of the part that he’s bringing to class, how much of the whole pan would that be? Write an equation to represent this, and then use a fraction bar or number line to explain your answer.
During the lesson planning process, Natalia, Tim, and Nicholas imagined multiple possible student responses and brainstormed questions they would ask to elicit their classmates’ mathematical reasoning: “I noticed that you first split the number line in half. Why did you do that?” “Is there another way that you can represent the problem?”
When it came time to teach the class, the three candidates circulated throughout the classroom, then selected classmates – each of whom had taken a different approach to representing the brownie scenario – to present their solution. “An integral part of any lesson plan,” explains Tim, “is allowing students to voice their opinions and thoughts and having other students respond, because those are the connections that help deepen conceptual understanding.” For a longer look at Tim leading a class discussion, click here.
To appreciate why it’s so important for preservice teachers to practice and see models of rigorous instructional tasks, consider Tim’s experience teaching the lesson. As an elementary-school student, he says, he learned the standard algorithm, but he doesn’t recall learning why the algorithm worked as it did. “I know I need to multiply straight across. I get that. Procedure, done,” he told his colleagues during the planning session.
By digging into the task – and having to facilitate a mathematical disagreement between his classmates – Tim encountered gaps in his own understanding. Is 1/3 times 1/2 the same thing as 1/2 times 1/3? Yes … if you’re just doing the procedure; both equations generate an answer of 1/6. (And by fifth grade, children have learned the commutative property of multiplication, which states that the order in which numbers are multiplied does not affect the product.)
But if you want students to be able to visualize a third of a half, then the order does matter, because a third of a half looks different from a half of a third. Are you starting with half of a brownie and icing a third, or do you have a third of a brownie and you’re icing half? “I want their verbal explanations to be able to match their visual representations,” says Dr. Reinke of his teacher-candidates. That’s when he knows that they have developed the necessary content knowledge for teaching.
Two days after the candidates had taught the lesson to their classmates, Dr. Reinke returned to the topic to address some common misconceptions and dig more deeply into the task. For a longer look at Dr. Reinke clarifying misconceptions, click here.
Beginning teachers aren’t going to be experts in every North Carolina math standard. But if they understand the big ideas that spiral throughout the standards, and use rigorous tasks to explore them, then they will be better able to unpack the content when they are hired to teach fifth-grade, or any grade.
Mathematical concepts connect to each other.
They can be represented in multiple ways.
They help us to explain why algorithms work.
These big ideas “would transfer to almost any mathematical concept that you can think of,” Reinke says. But if students experience low demand tasks, they never learn the concepts. And if future teachers are exposed only to low demand tasks, they are never able to access the big ideas.
Says Reinke: “If you choose a low quality task, you’re limiting the variety of student responses that you’re going to get. So you don’t have the chance to achieve a productive, interesting mathematical discussion because there are fewer perspectives that might be brought and students will have less to grapple with. You get a lot fewer disagreements or interesting conversations about those disagreements.”
Teaching future teachers to master big ideas and turn them into rigorous instructional tasks for children is a matter of educational equity. In our visits to K-12 classrooms and educator preparation programs across the country, we consistently see students – and future teachers – exposed to mindless instructional tasks, a “pretty useless waste of time,” as Dean McIntyre says.
All children deserve to be challenged and engaged in school. This is a matter of content. It is a question of tasks. And it requires teachers who focus deeply on their students’ thinking and recognize what kids are capable of.