Beyond the Common Core. Juli K. Dixon
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A unit of instruction connects topics in mathematics that are naturally grouped together—the essential ideas or content standard clusters. Each unit should consist of about four to six essential learning standards taught to every student in the course. These essential learning standards may consist of several daily learning objectives, sometimes described as the essential questions that support your daily lessons. The context of the lesson is the driving force for the entire lesson-design process. Each lesson context centers on clarity of the mathematical content and the processes for student learning.
The crux of any successful mathematics lesson rests on your collaborative team identifying and determining the daily learning objectives that align with the essential learning standards for the unit. Although you might develop daily learning objectives for each lesson as part of curriculum writing or review, your collaborative team should take time during lesson-design discussions to make sense of the essential learning standards for the unit and to consider how they are connected. This involves unpacking the mathematics content as well as the Mathematical Practices or processes each student will engage in as he or she learns the mathematics of the unit. Unpacking, in this case, means making sense of the mathematics listed in the standard, making sense of how the content connects to content learned in other grades as well as within the grade, and making sense of how students might develop both conceptual understanding and procedural skill with the mathematics listed in the standard.
The How
As you and your collaborative team unpack the mathematics content standards (the essential learning standards) for the unit, it is also important to decide which Standards for Mathematical Practice (or process) will receive focused development throughout the unit of instruction.
Unpacking a Learning Standard
How can your team explore the general unpacking of content and linking the content to student Mathematical Practices for any unit? Consider the third-grade mathematics content standard cluster Understand properties of multiplication and the relationship between multiplication and division in the domain Operations and Algebraic Thinking (3.OA). This content standard cluster consists of two standards: “Apply properties of operations as strategies to multiply and divide” and “Understand division as an unknown-factor problem” (NGA & CCSSO, 2010, p. 23). For the purpose of this discussion, focus your understanding on how to apply strategies based on properties of operations to multiply. See also the CCSS website (www.corestandards.org) to explore the Mathematical Practices and standards.
You can use the discussion questions from figure 1.2 to discuss an appropriate learning process you and your collaborative team could create for applying strategies to multiply.
Figure 1.2: Sample essential learning standard discussion tool.
Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.
Think about how you would expect students to demonstrate their understanding of both the standard of using properties correctly and Mathematical Practice 7, “Look for and make use of structure.”
Share your property-based strategies to find 6 × 7 with your team members, and discuss which properties you used to find the product. For example, you may have created a web to illustrate the properties you used (such as figure 1.3, page 12). Once your team brainstorms various strategies, focus your team discussion on the connection between the strategies and properties of operations.
Figure 1.3: Strategies for finding 6 × 7.
Your collaborative team’s conversation might have been similar to what follows: “What strategies can we use to find 6 × 7?” One collaborative team member might say, “I think of 6 × 7 as 3 × 7, and then I double that product.” Another might use the break-apart strategy and say, “I break apart the 6, and I think 5 × 7, and then I add 7.” The team should continue in this manner sharing strategies for 6 × 7. Eventually, the conversation will need to connect strategies to properties of operations if all students are to achieve the learning standard.
The team should identify properties including the commutative property, associative property, and distributive property. How can your team apply those properties to support the strategies identified for 6 × 7? Exploration of the mathematics standards at this grain size is crucial for making sense of the learning standards and should take place in the collaborative team setting. Team members must feel comfortable exploring the mathematics they will teach and discussing uncertainties within the collaborative team.
This is especially important since you or your colleagues were not necessarily taught the way you are expected to teach using various strategies that develop understanding. For example, a team member might not know what property connects to the doubling strategy in figure 1.3. Others on your team should help this teacher understand how the associative property supports this strategy because the teacher is thinking of the 6 as 2 × 3 and rather than thinking about (2 × 3) × 7, the teacher is using the strategy 2 × (3 × 7).
Similarly, the team can connect the distributive property to the break-apart strategy we described earlier. Together you might see that when using the break-apart strategy to think of 6 as 5 + 1, students have 6 × 7 = (5 + 1) × 7, and they are using the distributive property to multiply 5 × 7 then adding 1 × 7, because 6 × 7 = (5 + 1) × 7 = (5 × 7) + (1 × 7).
This level of unpacking of the essential learning standards is crucial before you can plan for effective student engagement with the mathematics. For example, once you make sense of how to use the properties of operations as strategies to multiply, it might become more clear that students could engage in Mathematical Practice 7, “Look for and make use of structure,” to explore how strategies based on properties of operations can help when multiplying.
You might also observe an application of Mathematical Practice 8, “Look for and express regularity in repeated reasoning,” when planning opportunities for students to use the doubling strategy to solve multiplication problems that have even factors by providing several examples where that strategy is useful.
Unpacking a Unit
The key elements of this first high-leverage team action are making sense of the essential learning standards, planning for student engagement in the