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A DEEPER LOOK INTO PROGRESSIONS Team Task Choose 1 Progression to READ: 1. 3-5 Number and Operations--Fractions 2. 6-7 Ratios and Proportional Relationships Develop “Content” Knowledge Package Knowledge Package BIG IDEAS that are foundational for understanding Facts or procedures that help in solving problems fluently and efficiently Visual models that make concepts and relationships most clearly visible Approaches for accomplishing a task that help make sense of the mathematics Vocabulary, terminology, or phrases that are necessary for participation in discussion about the topic Unpacking YOUR Knowledge 169 ÷ 14 Your homework was to solve the division problem using two strategies other than the conventional algorithm. You explained and represented your thinking using symbols, words, and diagrams, as appropriate for each strategy. Unpacking Division Reflections 4 Square Reflection share out “Knowing mathematics sufficiently for teaching requires being able to unpack ideas and make them accessible as they are first encountered by the learner.” -Ball, 2003 READ and REFLECT Liping Ma’s Knowledge Packages Liping Ma used the term “knowledge package” when speaking about the subject matter knowledge of teachers. She explains that when a teacher begins to teach a new topic, that teacher has an idea in her mind about where this idea is situated in the field of mathematics. Thus, “given a topic, a teacher tends to see other topics related to its learning,” and such topics comprise the knowledge package for the topic to be taught. In knowledge packages, there are “key pieces,” which are certain related topics that are viewed as being more important to the comprehension of the topic at hand. Knowledge packages for any topic can contain both procedural and conceptual elements, and Ma asserts that the two are interrelated. Ma found that teachers with a conceptual understanding of a topic viewed related procedural topics as being essential to student understanding. “In fact,” Ma emphasizes, these teachers felt that “a conceptual understanding is never separate from the corresponding procedures where the understanding ‘lives’. Ma believes that knowledge packages are important because it is from this information that a teacher attempts to construct a cohesive and comprehensive picture of a mathematical topic. With underdeveloped knowledge packages, it can be very difficult for a teacher to plan and facilitate a course of study for their students. Learning Paths – a linear progression? Knowledge packages also contain implicit sequences of student learning, where a student is expected to know and understand key related pieces before they can grasp the topic at hand. Ma explains, “the teachers believe that these sequences are the main paths through which knowledge and skill about the… topic develop” . While this may seem to be a linear progression of learning (you need to know x, y, and z, before you can learn topic A), Ma clarifies that topics in a knowledge package are interdependent, and that “linear sequences, however, do not develop alone, but are supported by other topics”. Thus, the learning paths generated through teachers’ knowledge packages are similar to the learning trajectories. Knowledge Package Structure Organizing our thinking about a mathematics topic around the Knowledge Package, provides us with a structure for asking these same five questions for all mathematics topics. These categories allow us to understand what is important for students to know, to teach with clarity and intentionality, and then to assess student understanding with minimal ambiguity. Knowledge packages provide an important pattern for learning and teaching mathematics. Multiplication Knowledge Package – 3rd Grade Knowledge Packages include Five Key Components: 1. Conceptual Understanding: Big overarching ideas that are foundational for understanding 2. Skills: Facts or procedures that help in solving problems fluently and efficiently 3. Representations: Visual models that make concepts and relationships most clearly visible 4. Strategies: Approaches for accomplishing a task that help to make sense of the mathematics 5. Mathematical Language: Mathematics vocabulary, terminology, or phrases that are necessary for participation in discussions about the topic Division Package of Knowledge Teams: Unpack knowledge needed to solve this problem Whole Group: Record team responses on Chart Paper TEAM TASK: Record “knowledge responses” on post-its Sort “Knowledge” into 5 Key Components Create “Knowledge Package” (justify and defend) Purposes of Knowledge Packages Liping Ma’s “knowledge packages” provide us with a useful way to clarify the ideas that students must understand in mathematics content. Mathematics standards are typically listed in linear form. They tend to reduce mathematics to disconnected lists of knowledge and skills. In contrast, knowledge packages highlight connections between and among ideas. Knowledge packages provide the context and the glue for mathematics standards, bringing ideas together into a more comprehensible whole. Knowledge packages may address a large topic, such as multiplication, or describe a task within a topic, such as comparing fractions. The specific contents of a knowledge package vary according to the grade or developmental level being addressed. Using Knowledge Packages Knowledge packages are not static words on a page, but instead can be the focus of discussions as teachers discuss relationships between standards, prioritize ideas, and seek out new and better ways to teach their students. It is the connections among these ideas, concepts, and procedures that enable teachers to portray mathematics as a unified body of knowledge rather than as isolated topics. (Ma 1999) Knowledge Packages: Simple Tools of Change The simplicity of Knowledge Packages allows schools to turn on a dime and realign their thinking about mathematics. When teachers come to understand the relationships within knowledge packages, they find their voices. They learn what is important for students to understand, and they ask the right questions about how concepts, skills, strategies, representations, and mathematical language relate. When educators throughout the school ask these same questions, student experiences from class to class, from year to year, connect. Everyone is a link in the chain, and it takes time for links to forge. That forging happens when educators gain outside support and when they engage in school-based research about knowledge packages. Knowledge Packages are the chains that bind our thinking about mathematics and the test of their strength is student success. CREATING YOUR OWN KNOWLEDGE PACKAGE IF THE HAMMER IS THE ONLY TOOL YOU GIVE STUDENTS . . . . EVERYTHING BEGINS TO LOOK LIKE A NAIL!