unit, course, or grade level?
2. How will we know when each student has acquired the essential mathematics knowledge and skills?
3. How will we respond when some students of mathematics do not learn?
4. How will we extend the learning for students of mathematics who are already proficient?
The four critical questions of a PLC at Work provide an equitable formative process for your professional work in mathematics assessment, intervention, instruction, homework, and grading. Imagine the access and opportunity gaps that will exist if you and your colleagues do not agree on the core learning standards for each unit as well as the level of rigor for the essential question (question 1): What do we want all students to know and be able to do?
Imagine the devastating effects on students if you do not reach team agreement on the rigor of lower- and higher-level cognitive demand for the mathematical tasks you use to engage students in mathematics lessons and assessments (question 2).
Imagine the lack of student agency (voice, ownership, perseverance, and action during learning) if you do not work together to create a unified, robust formative process for helping students own their response during class, reflecting when they are and are not learning during the lesson and after each assessment (questions 3 and 4).
For these reasons, we refer to our process as Mathematics in a PLC at Work. For you and your colleagues to effectively answer the four critical questions of a PLC at Work, in regard to a lesson’s instruction and tasks, requires the development, use, and understanding of lesson-design criteria that cause students to engage in the lesson, persevere through the lesson, and embrace their errors as they demonstrate learning pathways for the various mathematics tasks you present to them. Additionally, answering the critical questions well while planning for homework, grading, assessment, and intervention requires structure through the development of products for a team’s work together. It also requires a formative culture through the process of how you work with your team to use those products.
Your team reflecting together and then taking action around the right mathematics lesson-design work is the key to improved student learning. The actions you and your colleagues take together can improve the likelihood of more equitable mathematics learning experiences for every K–12 student. The reflect, refine, and act cycle illustrates this perspective about the process of lifelong learning.
This is a formative learning cycle. When you embrace mathematics learning as a process, you and your students:
• Reflect—Work the task or tasks, and then ask, “Is this the best solution strategy?”
• Refine—Receive FAST feedback and ask, “Do I embrace my errors?”
• Act—Persevere and ask, “Do I seek to understand my own learning?”
This cycle provides a systematic way to structure and facilitate in-depth team discussions. Additionally, the Mathematics in a PLC at Work framework on the next page focuses on six teacher team actions and two mathematics coaching actions within four primary categories. The eight actions focus on the teacher teams’ professional work and how they should respond to the four critical questions of a PLC at Work (DuFour et al., 2016).
The reflect, refine, and act cycle and the Mathematics in a PLC at Work framework will guide your work in planning unit assessments, daily instruction, daily homework, grading practices, and systematic interventions.
Mathematics in a PLC at Work Framework
Your Work and Your Story
Your work as a teacher of mathematics tells a story over time. That story, steeped in the decisions you make year after year, eventually becomes your career. Along the way, you must decide which parts of your journey are the most important to pursue so that your daily effort and toil can make a difference in the mathematics learning of the students you teach.
Gaining clarity on your vision for mathematics teaching improvement will result in helping your students achieve greater agency and ownership over their learning as the school year progresses. One of the best benefits of working in a community with peers is the benefit of belonging to something larger than yourself. There is a benefit to learning about mathematics from each other, as professionals. It is often in a community that we all find a deeper meaning to our work and strength in the journey as we solve the complex mathematics learning issues we face each week of the school calendar, together.
Thus the mathematics assessment and intervention, instruction and tasks, and homework and grading vision of the Every Student Can Learn Mathematics series features a wide range of research-affirmed voices, tools, and discussion protocols that offer advice, tips, and knowledge for your PLC at Work–based collaborative mathematics team.
As a teacher and leader of mathematics, your daily work and actions tell a story. That story reveals itself through your collaborative actions with colleagues around three important aspects of your daily work.
The Story of Your Mathematics Assessment Design and Intervention Routines
Your successful assessment story uses the essential standards for each mathematics unit to drive the assessment process in your school and uses those assessments for effective mathematics intervention. The research of James Popham (2011) and others highlights highly effective mathematics assessments designed to expect student participation in a reflect, refine, and act cycle of learning; and more important, to support students to take ownership of the learning process.
Although this may sound complicated, it is primarily a matter of refining your current mathematics assessment story and effort into a more efficient routine that looks something like this: design high-quality mathematics assessments, score (grade) samples of student work on those assessments together, pass quizzes and tests back to your students for an analysis and response to errors (students reflect and refine), and require your students to take action on the standards they have not yet mastered (students embrace and then act on their errors).
The Story of Your Mathematics Instruction Design and Lesson Routines
Through your experiences as a mathematics teacher, and your deep dive into the research on how students learn mathematics, you examine closely the educational research in the mathematics profession and find the right criteria for K–12 mathematics lesson design.
The elements of effective instruction are certain. Yet, those criteria are not prescriptive. That is, the research provides the freedom to act and teach mathematics within the well-defined boundaries of those criteria. As a mathematics teacher, leader, or school principal, you lead the way in describing how a student formatively reflects, refines, and acts when using the lesson-design criteria. In doing so, you will discover that the most effective K–12 mathematics lessons present a story of student perseverance and engagement during the lesson.
That story includes great lesson openings through prior knowledge and vocabulary work development. The story then moves to great lesson development using tasks that show a balance of lower- and higher-level cognitive demand and a balance of whole-group and small-group discourse as part of a sustainable formative-feedback process. The story ends with great lesson closures that students lead. Can the students provide evidence
