UNPACKING THE MATHEMATICS FOR DEEPER CONCEPTUAL LEARNING-CONCEPT MAPPING
In our lesson study planning, we work with our teachers to identify a content standard or topic that presents a problem of practice. Typically, teacher teams select a topic in their curriculum that presents a challenge in their instruction or presents a gap in students understanding as revealed by their pre-assessment or benchmark assessments. Once the team selects the content strand that they want to focus on for their lesson study, the team spends some time delving into the content standard by unpacking the mathematics using a concept map. Similar to Liping Ma’s (1999) “knowledge package,” the concept map is a way to unpack teachers’ understanding about the subject- matter knowledge of the concept.
Knowledge packages for any topic can contain both procedural and conceptual elements, and map out the concepts that are prerequisite and future mathematics concepts that build upon one another. Ma found that teachers with a conceptual understanding of a topic viewed related procedural topics as being essential to student understanding. Before we start a lesson study, we ask teachers to map out the conceptual understanding of a topic. We start with the concept at the center of the concept map then from there, map out and link the big ideas that are foundational skills and prerequisite to the core concept and the related skills, procedures, and strategies that are critical to developing fluency and efficiency.
Next, teachers map out and name the various visual representations that students see in their everyday life or in school such as manipulatives and models. In addition, teachers identify the important mathematics vocabulary, relationships between concepts that is essential in providing access to mathematics, and help in the mathematics discourse (see Figure 2.1). This unpacking process provides opportunities for teachers to engage in discussion about the vertical learning progressions as teachers from different grade levels discuss the learning objectives specific to their grades.
It also provides an opportune time for a math coach or the professional leader to introduce teachers to relevant research-based articles and materials for teachers to consider. This also marks the beginning of the lesson study cycle often referred to the “study” cycle because teachers bring to the table the problem of practice or an area in their curriculum that seems most troublesome to teach and/or learn and gather all their curricular materials and lessons that they have used in the past for close examination.
Figure 2.1 Unpacking the concept of relating decimals and fractions.