Beyond the Common Core. Juli K. Dixon
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 Mathematical Practices or processes that support them, and deciding on common pacing for the unit. These elements need to occur before the unit begins in order to take full advantage of instructional time during the unit. In the case of the content standard cluster Understand properties of multiplication and the relationship between multiplication and division in the domain 3.OA, without teacher team focus on unpacking the learning standard, students might not be urged to move past drawing pictures of groups of objects to multiply. Instruction might be limited to moving directly from drawing pictures of groups to memorizing basic multiplication facts. Both of these options do not meet the learning standard “Apply properties of operations as strategies to multiply.”
Your collaborative team may need to use outside resources to make sense of the mathematics involved in the unit. The background information in your school textbook teacher’s edition and digital resources can be a good source for this foundational knowledge, as can resources from the National Council of Teachers of Mathematics (www.nctm.org), such as the Essential Understanding series.
In general, your team can use figure 1.4 (page 14) as a planning discussion tool to help you better understand the essential learning standards in each of your grade-level units.
Thus, unpacking your own understanding of the essential learning standards, narrowing the more specific daily learning objectives, and identifying appropriate Mathematical Practices and processes to support those standards are crucial steps to providing a clear path to impact student learning. This will ensure your students will benefit from opportunities for deeper understanding during the unit.
Consider the sample unit plan for grade 3 in figure 1.5 (pages 15–17) to help students develop strategies based on properties of operations to multiply (supportive of domain 3.OA and essential learning standard 3.OA.5).
Figure 1.4: Discussion tool for making sense of the agreed-on essential learning standards for the unit.
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Figure 1.5: Sample unit plan progression of content for applying properties of operations as strategies to multiply for grade 3.
*While some Mathematical Practices are pervasive throughout the unit, such as Mathematical Practice 3, it is important to target specific practices for planning purposes.
Source for standards: NGA & CCSSO, 2010, pp. 6–8, 23.
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Notice how figure 1.5 also provides guidance for the common pacing expectations of the unit. While unpacking the essential learning standards, your team will need to reach agreement on the total number of days needed for the unit, the expected date for the end-of-unit assessment, and the timing of your review for student performance on the end-of-unit assessment (discussed further in chapter 3, page 121).
Your Team’s Progress
It is helpful to diagnose your team’s current reality and action prior to launching the unit. Ask each team member to individually assess your team on the first high-leverage team action using the status check tool in table 1.1. Discuss your perception of your team’s progress on making sense of the agreed-on essential learning standards and pacing. It matters less which stage your team is at and more that you and your team members are committed to working together to focus on understanding the learning standards and the best activities and strategies for increasing student understanding and achievement as your team seeks stage IV—sustaining.
Table 1.1: Before-the-Unit Status Check Tool for HLTA 1—Making Sense of the Agreed-On Essential Learning Standards (Content and Practices) and Pacing
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