Providing Teachers with a Flexible Base of Knowledge to Help Struggling Students
“‘We have to get better. And we can...we can add a layer.’ That layer will remain reaching individual teachers and supporting them as they assume full ownership for enacting Common Core State Standards (CCSS) in their classrooms.” Our recent report, Teaming for Success: Common Core Implementation at an Urban High School, highlighted the final step in implementing the Common Core, that of individual teacher ownership, through documenting the challenges and successes experienced by the High School of Telecommunications Arts and Technology (Telly) during their four-year rollout of the CCSS. Telly, a high-performing high school in Brooklyn, made department and grade teams the focal point for their rollout strategy, knowing that in order to achieve the deep, authentic change in teaching that the Common Core requires, teachers would need the time, space and structure to collaboratively learn, develop and share their experiences implementing the new standards.
This focus on building the capacity of the team to meet the goals set out by the CCSS is echoed in the work of the New Visions program, Accessing Algebra Through Inquiry (a2i). Using collaborative inquiry, a2i instructional specialists work with teams of math teachers in 30 New Visions schools, building their capacity to implement complex curricula, uncover students mathematical thinking and plan for instruction based on their understanding of student needs. A2i provides teachers in the program with rich mathematical tasks that are specifically designed to gauge students’ mathematical understanding. A2i instructional specialists work closely with individual teachers and teacher teams to develop opportunities for students to share, develop and extend their understanding.
A complex endeavor
However, as we discovered in the study of Telly, achieving the vision set out by the Common Core is a complex endeavor. Based on Common Core requirements, students must demonstrate both an understanding of content and proficiency in mathematical procedures, teachers must support “a balanced combination of procedure and [conceptual] understanding…[as] students who lack understanding of a topic may rely on procedures too heavily.” In other words, while a student might understand how to construct a perpendicular bisector between two points, without sufficient conceptual knowledge associated with this procedure, they would not know to apply it to the following problem: Alberto and Betty both heard thunder outside at the same time. Using the diagram that shows their positions, indicate all of the possible places the lightning could have struck.
For teachers, this means their task, is in providing “a flexible base [of content and skill knowledge] from which [students] can...connect practices to the content.” By understanding the earlier grades’ standards, or the foundation for the “flexible base,” high school teachers are better able to draw connections between and among the concepts represented in the Common Core State Standards and provide students with authentic opportunities for the application of these concepts. Without this understanding, however, teachers are less able to help students who have fallen behind grade level as they lack the understanding of the foundational skills that should have been mastered in previous grades. Susan Fairchild discusses the system dynamics that produce these off-track students, as well as the implications of maintaining these systems within schools, in a recent blog post for EdWeek.
Applying a "flexible base" to geometry
Recognizing the challenge this poses to teachers, David Wees, a formative assessment specialist for the a2i program, led a session at a recent cross-network professional development convening, and began this work with geometry teachers. David structured the intended product of the session on Jason Zimba’s diagram of the Standards for Mathematical Content, which illustrates the connections between the content standards and spans kindergarten through eighth grade. The diagram provides a visual of the narrative learning progression documents that formed the basis for the Common Core State Standards of Mathematics (CCSSM). Because this type of diagram does not exist, at least publicly, for high school Common Core-aligned Geometry standards, David imagined that this work would be valuable both for teachers participating in the session and potentially for a more national audience.
To begin the session, teachers individually completed a multi-step math problem, with many possible pathways to the solution which was aligned to the mathematics of the upcoming unit (all a2i teachers follow a common scope and sequence). After a brief discussion of the various approaches to the problem, David posed a question to the group to frame the work of the session: “How do we identify kids struggling with prerequisite skills and knowledge that aren’t directly connected to or assessed in the tasks for this unit?” To begin to answer this question, David divided the room into groups by grade-level standards for grade 5 through high school (including algebra). These groups examined each standard within their assigned grade level and discussed whether or not they represented the mathematic skills necessary to be successful in the upcoming unit, Unit 4. Once they had separated the standards that were related to Unit 4 from those that were not, each group placed these standards on the chart pictured above, drawing arrows and linking each standard together with others based on their hypotheses for how they were interrelated. After individual groups had completed this charting exercise for their distinct grade, groups teamed up to connect standards between grades.
Seeing the big picture
While the diagram that emerged from this session was much narrower in scope than Zimba’s, the value for participants was apparent. Teachers uniformly acknowledged that this activity allowed them to see the connections between this unit’s standards and the standards of previous grades. In addition, most teachers shared that they wanted to return to this work with their departments in planning for the enactment of the units. Telly teachers, in particular, stressed that diagramming would be useful in teaching certain algebra skills, such as solving quadratic equations. They realized that because students were no longer taking algebra before geometry at their school, they needed to be particularly attentive to the prerequisite algebra skills for each of these geometry units.
The end product: a diagram of the prerequisite skills, beginning in 5th grade, for the core mathematical standards of a2i Geometry Unit 4.
Mr. Wees believed that the session led to two concrete outcomes: 1) every participant had the opportunity to read through at least one grade of elementary or middle school standards, which most teachers had not done before; 2) all participants walked away with the knowledge that these prerequisite skills are important for students’ current knowledge. David hopes that teachers will now be more likely to connect students’ struggling with a lack of prior knowledge and work to identify what prior knowledge is missing. Much of the ongoing work for the a2i instructional specialists is building teachers’ pedagogical content knowledge so that they are better able to address these prerequisite skills gaps with their students.
Since this exercise, David has slightly enhanced and digitized the teacher-produced diagramn (seen below). David took the existing networking of standards from Jason Zimba as the starting place for the k-8 piece of the visualization, and built up into high school based on the teachers’ work during the session. He then vetted the high school work and made minor changes where the diagram diverged from his understanding and experience of the standards.
Given the overwhelmingly positive response to this session, the a2i team is determining next steps. Many teachers shared that they hope to continue this work with each of the subsequent units, even potentially going back to units 1, 2, and 3 for a more developed understanding of the course sequence as a whole. Long term, the team envisions designing a set of mathematical tasks that follow along a specific trajectory to the high-school level standard; this way, teachers have a clearer sense of how they might support their students in building up their “flexible base” of mathematical skills and content knowledge.