Perseverance and Cultures of Encouragement
As our school continues to plumb the depths of the Common Core and discover new meanings and classroom trajectories for implementation, we often return to the Mathematical Practices as an articulation of a larger vision for what students should know and be able to do in the math classroom. The defining of a robust set of practice standards that are held broadly across states for all students is one of the critical contributions the standards have made for math education at the policy level. It serves as a clear recognition that while learning specific mathematical concepts and skills is vitally important, it is equally important to develop habits of mind or mathematical practices that we foster or encourage in our students.
The first mathematical practice, Make Sense of Problems and Persevere in Solving Them, is one that I am keen on witnessing and fostering in our math classrooms. I am specifically interested in perseverance, and how the math classroom encourages students to not give up, to believe in what they can do, tackle obstacles, recognize “failure” as part of learning, and find their way to an outcome. As the posters above and the title suggest, I saw a teacher who is developing a culture of encouragement that has led to progress on this mathematical practice. I see her weaving a way between two commonly used approaches to address perseverance and problem-solving.
Isle of Perseverance
The conversation on mathematical practice 1 often turns to how we can develop perseverance “in” our students. As a result, perseverance becomes an individual problem or issue. In this individualistic vein, teachers focus on students using multiple representations (visual models, paraphrasing, patterns, tables, graphs, manipulatives, etc.) or on developing a checklist or problem-solving process (Example 1, Example 2, Example 3 …). It is critically important for students to think of problem-solving as a marshaling of a diverse set of tools and skills as well as a thoughtful process of chipping away at a problem. This necessary approach supports perseverance by developing internal resources to tackle a task or problem. While this often leads to the adoption and use of a broad set of skills, the tough-to-capture disposition of perseverance or the often essential movement to turn to another for advice or inspiration is often excluded.
Teachers who move off the isle of perseverance often turn problem-solving over to the dynamics of partners and groups so that students can support each other in problem-solving. Through questions or structures teachers often turn students back toward each other and attempt to foster dialogue among themselves to find new ways forward in solving a problem. Teachers and students alike must listen, speak, build off the ideas of others, and rely on each other to negotiate the given problem at hand (excellent examples can be found at the Mathematics Assessment Project Lessons). This too is a necessary approach to both sustaining the problem-solving approach as well as developing the habits of working together and learning from each other; however, the question of whether a student can transfer these lessons learned to a moment where they are solving by themselves often proves difficult.
Cultures of Encouragement
Bridget does both of the things above in her classroom, but I also witnessed a middle road that she carved between more individualistic approaches and approaches that are more group-based. The video below is what I was fortunate to see (it also seemed to be one embodiment of what students wanted from their class as indicated in the pictures of the posters at the top of the post):
She huddles the whole class together with each student as a possible participant. One student emerges to determine the location of a number on a vertical number line and justify/explain her placement. The students does this by herself, but before her peers, and the class acts like a team. They offer her encouragement on her effort and explanation with snaps of the fingers. Others, in dialogue with the teacher, come to an accurate solution. The student then she comes forward and is praise for the bravery she showed in correcting her initial answer. All the students, again, offer her encouragement with the snaps of their fingers.
Without trying to narrowly box this experience or generalize to broadly from it, I took three things from it:
While aspects of perseverance in problem-solving can be developed individually and in small groups, building moments where the whole class is functioning as a community can serve as a resource for perseverance. In the clip, Bridget created a whole class by moving them close together, almost as a huddle, and by making the event of learning an intimate, common experience. While the student in the video had to answer the problem by herself, she was not alone. Other classmates were invited to step out and offer their voice after her initial attempt. The whole class was sharing the problem. As a result, the learning happens through the class (writ large), and the individual and class work together.
2. Student Praise
When students are working individually they are often praised by the teacher. When students are in groups, they may praise or encourage each other. Rarely, however, is a student praised by the entire class. The praise was not only communal, but more importantly, it was praise directed at her attempt to solve not at the answer achieved. This builds the class a whole around a culture of encouragement, and also builds the individual as well.
The turn from the individual back to the whole class to think about the work and offer revisions maintained a balance between the process and attempt to solve with achieving an accurate solution. The entire class was involved in the revision process and worked toward an accurate solution; revision did not stop there, but returned back to the student whose work they revised. The initial student stepped back to the fore, bravely (as Bridget stated before the class), and completed the revision herself. It is difficult to support perseverance and problem-solving without allowing for multiple attempts, learning from others, revising one’s first step, and possibly most importantly, being immersed in a culture of encouragement.
Mathematical Practice 1 is complex, difficult, and multi-faceted standard to meet within any classroom, but this brief look into a classroom convinced me that creating cultures of encouragement is one place to start.