Welcome to 5 Days of Exemplars, our deep dive into using rich performance tasks to build students’ problem-solving skills. We’ve already discussed how to support students as they unpack a problem and choose strategies to solve it. Now they’re equipped to engage with a challenging new task! Today, we’ll consider your role in supporting them in ways that empower them to discover solutions for themselves.

Solving an Exemplars problem-solving task starts with establishing conceptual understanding. Conceptual understanding asks students to make sense of the problem for themselves, to see the connections between the problem and their own internal, prior knowledge. How can a student utilize what they’ve learned and practiced in the past to help them begin to solve this new, unfamiliar task? 

The CRA Model

Within conceptual understanding there is a powerful protocol we can utilize with our students that’s defined as the Concrete - Representational - Abstract (CRA) model. 

This process begins by asking students to use hands-on manipulatives to explore the mathematical concepts in the task. Could a student count out a series of manipulatives that is described as the starting quantity in the problem using ten frames or Base 10 blocks? Could a student count using a rekenrek or snap cubes? Would fraction tiles or fraction circles help a student visualize the quantities in the task? Interaction with these concrete manipulatives allows students to explore a task within a safe and tangible way. Engaging classrooms should have manipulatives available around the room ( or virtually) for students to self-select for whatever math challenge they are working on. 

As students make sense of the problem, they can progress to utilizing representations. What makes an effective representation progresses through the grades. Early grades may use hand-drawn diagrams. Number lines can become a powerful tool that evolves in complexity from kindergarten all the way through middle school, high school and even graduate level calculus. Can the student create a number line on their paper or use an online virtual number line? If a lot of information has been given in a task, how might a student utilize a table to organize it? What would be the correct labels for the columns and rows? Tables arrive in the first grade. Area models and arrays can be powerful tools as early as the second grade. 

The goal of utilizing the concrete manipulatives and representations is to help our students “see” the relationships between quantities in a problem-solving task in a way that can be explored and “played” with. 

The final step in the CRA model asks students to translate the understanding they’ve developed through the use of concrete manipulatives and representations into a symbolic visualization of the quantities and operations in the task. It is within this stage that students explore procedures, algorithms and computational fluency.

Exemplars tasks are expressly designed to make the most of these tools—to allow students to use manipulatives and representations to initiate an understanding of how to solve it. Digital manipulatives such as Brainingcamp, may be used to support this process in a remote or blended setting. Watch the video below to see digital manipulatives in action.

Student Collaboration

Beyond the CRA, student collaboration is another powerful approach for solving a problem-solving task. In virtual classrooms, breakout rooms create terrific potential for small groups of students (3 or 4) to discuss ideas for solving the problem, to build representations, to make sense of the challenges of the task and to work towards a collective solution.

Research shows dramatically more brain activity when students are working collaboratively versus students working in isolation. Some research indicates the difference is so potent that collaborative learning counts as a social activity. The article Learning and the Social Brain, published in Edutopia, states, “If the species is hardwired to work together, then our classrooms should continue to feature a healthy dose of activities that emphasize cooperation, teamwork, and peer-to-peer teaching.” 

Ask Probing Questions

An important role for teachers, as students work collectively to solve an Exemplars task, is to ask probing questions without giving too much information away. Here are some ways to have the students explain in their own words where they are currently:

• Can you tell me what you’re working on?
• What do you know so far?
• What have you done so far?
• Are there any areas/parts/sections you’re unsure about?
• Where are you stuck? 

The goal of this time for the teacher is not to tell your students how to solve the problem, or to give them significant hints to push them in the right direction. The goal for the teacher is to help students make sense of the problem right where they are in their process of solving it. So what does the student need to know? What questions can you ask to help them see for themselves what they might be missing?

In your questions, try not to accidentally “steal” the magic of the moment of understanding that we’re aiming for our students to discover. The agency our students develop as mathematicians occur when they feel the power of truly figuring things out for themselves. Your role is to be the guide. You know the right/correct answer as the teacher—so engage in questioning that will let your students discover a solution to the task themselves.

All of these ideas are part of the Blue section of Exemplars Problem Solving Process - Solve the Problem. With time to explore possible solutions, opportunities to discuss strategies with their peers, and support in the form of encouraging questions from their teachers, students can develop the competence and confidence to solve complex problems—and come out of the experience excited and engaged 

What’s next, now that your students have set out to solve the problem? We’ll explore how to help them develop viable arguments and explain how they solved it—in short, to show their work and communicate their solution. To try these approaches in your classroom starting today, sign up for a free trial of the Exemplars Library. You can also request a quote or speak to us about your needs—or support your teachers’ skills through Exemplars professional development.