8 Chapters
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Epilogue The Promise and Possibility of Improved Mathematics Learning

Chris Weber Solution Tree Press ePub

Let’s imagine and design an improved approach to application-based mathematics—to all students learning mathematics deeply and at high levels. What if there were no standards? What if there were no high-stakes testing programs? How would we design units of mathematics instruction? How would we teach? How would we use assessment to inform our future supports and to communicate to students where they are and what they need to do?

We would start by nurturing a growth mindset among staff and students (Dweck, 2008). We would ensure positive and high expectations for students’ abilities to learn at high levels. We would truly believe in our ability to ensure that all students learn at high levels. We would trust our colleagues because no single teacher can meet all the needs of a student. We would nurture staff capacities and collaboration. A sense of collective responsibility is critical for all students learning mathematics.

For decades, we have not had a viable mathematics curriculum (Gonzales et al., 2008; Marzano, 2003; Schmidt, McKnight, Cogan, Jakwerth, & Houang, 1999). When the amount of content that teachers are attempting to cover is not viable or doable, some students fall behind, some students become frustrated, some fail, and many lack opportunities to learn deeply, so that they can apply and retain knowledge. We must prioritize the most essential mathematics topics that students will need to apply in real-world situations. Students must understand these conceptually and procedurally so that they are ready for the next grade level or course, and ultimately, for college or a skilled career. We must also ensure that all staff members have a common interpretation of what it will look and sound like when students demonstrate mastery of the prioritized topics. What is the level of rigor, and what is the format in which students will demonstrate mastery? This amount of focus will lead to clearly articulated sequences of content progressions, both horizontally (within a school year) and vertically (from year to year). In designing a successful, robust, and balanced mathematics program, clarifying and concentrating instruction is foundational.

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Chapter 4 Instructional Practices for Application-Based Mathematical Learning

Chris Weber Solution Tree Press ePub

Teaching mathematics is arguably one of the most complex elements of an elementary teacher’s profession. The nature of mathematical learning is such that students must master specific skills with a fluid understanding that allows them to apply the learning in a variety of contexts and in conjunction with other skills and understandings. In addition, students must master the requisite language and tools in order to be able to communicate and model mathematics. To design and implement instruction that ensures such rich learning, teachers must be able to weave evidence-based instructional strategies with mathematical practices and apply those elements strategically within engaging contexts.

Effective instruction includes those instructional decisions that positively impact student learning and engagement. The NMAP (2008) identifies these practices as follows:

• Maintenance of the balance between student-centered and teacher-directed instruction

• Explicit instruction for students having mathematics difficulties

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Chapter 2 Designing Conceptual Units in Mathematics

Chris Weber Solution Tree Press ePub

This chapter is about what we want students to learn and how we will know if they learn it (DuFour et al., 2010). We build the case for common assessments as a vital part of the unit-design process and describe how to craft them. As the key lever of RTI, common assessments establish the target that students and staff are working toward and provide the evidence that can be used to extend learning for some and intervene with others. In the second half of this chapter, we provide an example unit for grade 3.

When designing units of instruction, our goal must be mastery, not coverage—depth of understanding, not breadth of topics addressed—and we will describe just such a process for building units of instruction. First, how should units be organized and sequenced?

We recommend that units be coherently organized, both horizontally within a grade level and vertically between grade levels. Since a sense of number is the basis for all mathematics, we recommend that standards and learning targets that build students’ sense of number be frontloaded within a grade level’s instructional year. Within grade levels, we recommend that topics such as addition and subtraction, graphing, and algebra be included within individual units and the units that are adjacent to them to allow for more continuity of teaching and learning and greater depth of study. Between grade levels, coherence of topics allows for collaboration between grade levels and Tier 2 intervention, which is particularly relevant in smaller schools where there are one or two teachers per grade level.

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Chapter 1 Prioritized Content in Mathematics

Chris Weber Solution Tree Press ePub

There is, perhaps, no greater obstacle to all students learning at the levels of depth and complexity necessary to graduate from high school ready for college or a skilled career than the overwhelmingly and inappropriately large number of standards that students are expected to master—so numerous, in fact, that teachers cannot even adequately cover them, let alone effectively teach them to mastery. Moreover, students are too often diagnosed with a learning disability because we have proceeded through the curriculum (or pacing guide or textbook) too quickly; we do not build in time for the remediation and reteaching that we know some students require. We do not focus our efforts on the most highly prioritized standards and ensure that students learn deeply, enduringly, and meaningfully (Lyon et al., 2011). In short, we move too quickly trying to cover too much.

We distinguish between prioritized standards and supporting standards. We must focus our content and curriculum, collaboratively determining which standards are must-knows (prioritized) and which standards are nice-to-knows (supporting). This does not mean that we won’t teach all standards; rather, it guarantees that all students will learn the prioritized, must-know standards. To those who suggest that all standards are important or that nonteachers can and should prioritize standards, we respectfully ask, “Have teachers not been prioritizing their favorite standards in isolation for decades? Has prioritization of content not clumsily occurred as school years conclude without reaching the ends of textbooks?” Other colleagues contend that curricular frameworks and district curriculum maps should suffice. But we ask, “Will teachers feel a sense of ownership if they do not participate in this process? Will they understand why standards were prioritized? Will they stay faithful to first ensuring that all students master the must-knows, or will teachers continue, as they have for decades, to determine their own priorities and preferences regarding what is taught in the privacy of their classrooms?” We have found that the simplest and most effective way to determine a prioritized standard is to collaborate with teachers from the next grade level. For example, second-grade teachers should engage in vertical-articulation discussions with third-grade teachers, asking, “For what mathematics topics must incoming third graders possess mastery?”

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Chapter 3 Understanding the Prioritized Standards

Chris Weber Solution Tree Press ePub

An effective RTI framework for mathematics requires a cohesive, aligned curriculum (prioritized standards) that is taught through engaging, evidence-based instructional strategies that advance student ownership of mathematical learning. We have identified the why and how for prioritizing standards (introduction and chapter 1) and proposed units of instruction (chapter 2). However, we know that in order for teacher collaborative teams to implement those planning structures effectively in the classroom, they must also have clarity regarding the vocabulary and conceptual understandings that underlie the instruction. In this chapter, we provide guidance on the vocabulary and concepts; the next chapter provides guidance in designing application-based instruction.

The NMAP (2008) notes, “Research on the relationship between teachers’ mathematical knowledge and students’ achievement confirms the importance of teachers’ content knowledge” (p. xxi). More specifically, teachers’ understanding of mathematical content impacts critical instructional decisions, including identifying problem sets, questioning techniques, and connecting mathematical concepts (Hiebert & Stigler, 2004; Hill et al., 2005). The effectiveness of Tier 1 classroom instruction is significantly impacted by teachers’ understanding of the concepts and skills they teach and the ways those concepts connect to prior and future learning. Our focus in this chapter is to provide an overview or basis of conceptual understandings for the prioritized standards.

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