Читать книгу Beyond the Common Core - Juli K. Dixon - Страница 10

HLTA 1: Making Sense of the Agreed-On Essential Learning Standards (Content and Practices) and Pacing

An excellent mathematics program includes curriculum that develops important mathematics along coherent learning progressions and develops connections among areas of mathematical study and between mathematics and the real world.

—Steven Leinwand, Daniel J. Brahier, DeAnn Huinker,

Robert Q. Berry III, Frederick L. Dillon, et al.

In most K–5 grade levels, there will be eight to ten mathematics units (or chapters) during the school year. These units may also consist of several learning modules depending on how your curriculum is structured. An ongoing challenge is for you and your team to determine how to best make sense of and develop understanding for each of the agreed-on essential learning standards within the mathematics unit.

The What

Recall there are four critical questions every collaborative team in a PLC asks and answers on an ongoing unit-by-unit basis.

1. What do we want all students to know and be able to do? (The essential learning standards)

2. How will we know if they know it? (The assessment instruments and tasks teams use)

3. How will we respond if they don’t know it? (Formative assessment processes for intervention)

4. How will we respond if they do know it? (Formative assessment processes for extension and enrichment)

= Fully addressed with high-leverage team action

This first high-leverage team action enhances clarity on the first PLC critical question for collaborative team learning: What do we want all students to know and be able to do? The essential learning standards for the unit—the guaranteed and viable mathematics curriculum—include what (clusters and standards) students will learn, when they will learn it (the pacing of the unit), and how they will learn it (often via standards such as the Common Core Standards for Mathematical Practice). The Standards for Mathematical Practice “describe varieties of expertise that mathematic educators at all levels should seek to develop in their students” (National Governors Association Center for Best Practices [NGA] & Council of Chief State School Officers [CCSSO], 2010, p. 6). Following are the eight Standards for Mathematical Practice, which we include in full in appendix A (page 149).

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning. (NGA & CCSSO, 2010, pp. 6–8)

While different school districts use many names for learning standards—learning goals, learning targets, learning objectives, and so on—this handbook references the broad mathematical concepts and understandings for the entire unit as essential learning standards. For more specific lesson-by-lesson daily outcomes, we use daily learning objectives or essential questions. We use the terms learning goals or learning targets to reference the outcome for student proficiency in each standard. The daily learning objectives and the tasks and activities representing those objectives help students understand the essential learning standards for the unit in order to demonstrate proficiency (outcomes) on these standards. The daily learning objectives articulate for students what they are to learn that day and at the same time provide insight for teachers on how to assess students on the essential learning standards at the end of the unit.

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 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.

Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this figure.

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.

Visit go.solution-tree.com/mathematicsatwork to download reproducible versions of this figure.

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

Visit go.solution-tree.com/mathematicsatwork to download a reproducible version of this table.

Your responses to table 1.1 will help you determine what you and your team are doing well with respect to your focus on essential learning standards and where you might need to place more attention before the unit begins.

Once your team unpacks and understands the essential learning standards you are ready to identify and prepare for higher-level-cognitive-demand mathematical tasks related to those essential learning standards. It is necessary to include tasks at varying levels of demand during instruction. The idea is to match the tasks and their cognitive demand to the essential learning standard expectations for the unit. Selecting mathematical tasks together is the topic of the second high-leverage team action, HLTA 2.