It has been a tradition in many mathematics classrooms to follow the direct instruction model, but with the emphasis of the Common Core State Standards on developing students’ mathematical fluency, using a method that requires students to develop their thinking individually and then collaboratively (in that order) is consequential. This is particularly the reason why integrating formative assessments into the sequence of lesson planning is essential to teaching mathematical fluency. This strategy reverses the gradual release of responsibility by having the teacher give the students a task to complete in class or as homework prior to the lesson, in the form of a pre-assessment. The reasoning for this is to provide the teacher with an understanding of the students’ use of mathematics and any areas of opportunity. On the day of the lesson, the teacher passes back the students’ work with feedback or questions provided on each student’s work, encouraging the students to engage in self-reflection, “When we use assessment for formative purposes, students should receive growth-producing feedback and have the opportunity to make adjustments to their work based on that feedback,” (Rutherford, 2009, p. 139). This allows the student to consider their reasoning for using the mathematics that they did. Then, the teacher divides the class into small groups and has them work collaboratively on developing a joint solution. After students have developed a joint solution, the teacher holds a whole-class discussion. Prior to holding the discussion, the teacher should circulate around the classroom and note the types of approaches that each group is using to solve the task. The teacher can use these notes to help guide the discussion and explore the methods used. If time is available, the teacher could pass out several sample responses to each group, allowing them to analyze the different approaches and compare them to their own approach. Then, the teacher could hold a whole-class discussion on the different approaches used.
Teaching mathematical fluency is not only about teaching content, but about teaching students how to use that content, how to think about that content, and how to apply that content to the real world. Collaborating with others offers an opportunity for students not only to interact with their peers, allowing them to develop their mathematical fluency through listening and speaking with other students, but also to engage the content at a deeper level, “The fact that students are actively exchanging, debating and negotiating ideas within their groups increases students’ interest in learning. Importantly, by engaging in discussion and taking responsibility for their learning, students are encouraged to become critical thinkers” (Dooly, 2008, p. 2). As students interact and engage their peers in this model, they become active agents of their own learning and experience learning at a whole new level, “Educational experiences that are active, social, contextual, engaging, and student-owned lead to deeper learning” (Cornell University, 2013). Because this strategy offers a student-centered approach that builds on collaboration, it is generally applicable to all content areas. However, in terms of mathematics, it offers students a unique approach to mathematical fluency by improving how they think about mathematics. Based on the nature of this strategy, it provides for an excellent means of formative assessment, “Formative assessments…first allows students to demonstrate their prior understandings and abilities in employing the mathematical practices, and then involves students in resolving their own difficulties and misconceptions through structured discussion” (Mathematics Assessment Resource Service, 2013).