*This blog post is the fourth in a series of four blog posts exploring the student experience of problem-based learning. The first three posts are available here: (1) “**How Do Students Perceive Problem-Based Learning?**” (2) “**Inviting Students to the Mathematics**” (3) “**Concrete Representations that Give Students a Way to Get Started**.”*

Okay, so the kids can get started and represent their thinking. But are they really learning? Am I really teaching? What are we doing here?

*By Max Ray-Riek*

In this series of blog posts, we’re exploring how students have to shift their thinking about what being a good math student looks and sounds like in order to be successful in a problem-based classroom. What did you notice about the representations and tools offered to students? Are your students using those tools to support their reasoning and problem-solving? What are you curious about now, as you think about what it takes for students to be successful in a problem-based classroom?

Here is a reminder of one model for problem-based instruction that we use at IM:

Think back to the beginning of this series when we categorized how students often learn to behave in math class. When materials present students with all the tools and strategies they need before posing any problems, it’s less likely that students will experience learning from a classmate’s unique approach. It’s more likely that students won’t value their own mis-starts and incomplete ideas, because a complete model is always available. It’s more likely that conversations will end after the complete, correct answer is shared and checked against the authority. If only one approach is demonstrated, there’s little reason to compare and build connections among approaches that lead to the same answer.

In a problem-based curriculum, students do much of their learning during whole-group synthesis of different solution approaches. In other words, a lot of the learning happens when the teacher helps students compare and reflect *after* they’ve worked on the problem for a while. In the previous blog post, we described that the teacher might call the whole class back together when the students are at various problem-solving stages: one group might have finished the original problem a while ago and is now trying to solve it an additional way, while another group has a wrong answer, and a third group is only halfway through the problem. Why might the teacher do that? Why wouldn’t the teacher work with each group to make sure they are getting through this problem correctly?

Most of the learning in a problem-based classroom happens when students compare their ideas to one another. Armed with this knowledge, we can be more comfortable facilitating this sort of robust, synthesizing conversation. The teacher does not have to correct the group with a wrong answer because they know the whole class will benefit from comparing that group’s thinking to their own, and will understand better why the right answer is right. The teacher doesn’t have to wait for each group to finish because they know that the students had opportunities during the launch and work time to develop a better understanding of the problem. The teacher can set a goal of everyone being ready to make connections between their ideas and their classmates’ ideas, even if they struggled with the problem.

But how does this learning from others happen? It happens when students have shifted from thinking that the learning happens when the teacher explains (at the beginning of class) or corrects (at the end of class). It happens when students believe that learning happens from actively comparing their ideas to others’ ideas. It happens when students try hard to learn new, more efficient or effective ways of thinking.

In order for students to learn this way, they need to:

- Be willing to share their ideas, right or wrong.
- Be willing to listen to and learn from others’ ideas, right or wrong.
- Be curious about the work of other students, even if they have already solved the problem one way.
- Actively work to make sense of what other people are saying, even if they are confused.
- Be willing to try someone else’s method and work until they feel confident in their reasoning to solve similar problems in the future.
- Listen for structure, patterns, representations, and methods, not just answers and steps.

These norms are very different from the implicit or explicit norms they may have picked up in other math classes. How do we help students make the necessary shifts?

Culture change often happens when we bring active attention to it. Some methods teachers have used include:

- Using talk moves that support students to listen to one another and share their thinking such as:
- Giving independent think time both when starting to work on a task and after students hear another student’s ideas or methods.
- Using think-pair-share or turn-and-talk opportunities. Students can practice what they might say to the whole group, or tell someone what they understand and don’t understand about something they just heard, or tell a partner what connections they see between the strategy on the board and their own thinking.
- Asking multiple students to revoice what they heard a classmate say. Note that student revoicing tends to help students listen to each other and their peers. Teacher revoicing has the opposite effect, and decreases students’ willingness to listen to peers.

- Giving kids explicit feedback on their participation in discussions, before they focus at all on the content of discussions (“Tyler, I notice that when Jada shared her thinking, you asked her a question to understand it better, and that helped you both improve your ideas.”)
- Writing down examples of students actively using the classroom norms, and displaying them during the discussion as another way to give feedback (sometimes called a “Participation Quiz”). For example, you might display a seating chart, and then when you overhear a student say, “Can you explain how you thought to…?” you would write that quote down next to their group as a positive example of participation.
- Having kids create a running and growing list of class norms, and then helping students to actively enforce them (such as “don’t make fun of people being wrong,” or “try drawing a picture first before you ask a teacher for help,” or “give people quiet think time before jumping in.”) One way to grow this list is when you catch students engaged in productive behaviors, naming them and asking, “Should we add this to our list?”

Problem-based curricula like Illustrative Mathematics’ work best when kids believe they can tackle problems on their own, when they have tools to share their reasoning (not just their steps), and when they are willing and able to learn by comparing their ideas to their classmates’ in a discussion focused and guided by the teacher. Those skills are not always honed in their previous math class, if teachers used a different kind of curriculum. Therefore, explicit culture change for students is as important as picking a great curriculum.

### Next Steps

What do you notice as big shifts for your students when you use a problem-based curriculum? What are you still curious about when you think about how students learn from solving problems? How will you explore your questions in your reading, teaching, and conversations with colleagues?

##### Max Ray-Riek

**Max Ray-Riek **is the Director of 6-12 Professional Development at Illustrative Mathematics and worked as a writer on the high school curriculum. Before coming to Illustrative Mathematics, Max worked for The Math Forum, focusing on fostering problem solving and making student thinking the center of math class. He is the lead author of *Powerful Problem Solving: Activities for Sense Making with the Math Practices. *Max is a former secondary mathematics teacher (and before that, preschool teacher) who finds the art and discipline of valuing each student’s ideas, and supporting all students to see themselves as people who can listen, connect, and act mathematically, to be the hardest, most rewarding, and most important work in the world. Max lives in Philadelphia with his wife, with whom he fosters puppies who will grow up to be trained to be service dogs.

Thank you very much for this series-capping look at helping students make the shift into problem-based learning. In my district and at my school I have not very much of a community to draw on as I make my own shift into problem-based facilitating. Reading the techniques some other teachers have used is invaluable to me as I give a first try with the Illustrative Mathematics Math 1 curriculum this semester.