# Results for: “Mathematics”

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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. See All Chapters |
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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 See All Chapters |
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## Introduction The Rationale for 21st Century Mathematics |
Chris Weber | Solution Tree Press | ePub | ||||

—Robert Moses, Civil Rights Leader While schools have embraced the response to intervention (RTI) model for reading and behavior, implementation of RTI for mathematics continues to lag (Buffum, Mattos, & Weber, 2009, 2010, 2012). Several factors may contribute to this lag in implementation for numeracy. First, we have valued written and spoken language abilities over mathematics. It is also not uncommon or unacceptable for adults, including elementary educators, to say, “I never liked mathematics as a student” or “I’m not really good at mathematics.” It is less likely, however, that an educator would comfortably state, “I never liked reading” or “I’ve never been a good reader.” In addition, schools’ hesitation with the implementation of tiered instruction for mathematics may be impacted by educators’ levels of confidence with mathematics, mathematics instruction, and intervention. Often, the teachers with whom we partner freely express feeling less confident teaching mathematics than they do teaching language arts, and they often tell us they feel less professionally satisfied with the mathematics instruction in their classrooms. This may result not only from teachers’ lack of confidence in their own conceptual understanding but also from lower levels of confidence in instructional and intervention practices for mathematics. When we ask educators to reflect on their own mathematical learning, their memories include extensive experiences with worksheets, textbook pages, timed assessments, and round-robin competitive games designed to practice automaticity. Story or word problems are often omitted. The reality is that many of us experienced mathematics instruction that was abstract, procedural, and computational. While elements of those instructional practices may continue to have some value, the overdependence on them has likely contributed to educators’ lack of confidence teaching mathematics. Adults may compute and apply formulas proficiently; however, many find the fluid application and interconnected strategies of mathematics challenging simply because those elements have not traditionally been emphasized in classroom instruction—we are products of the very system we want to reform (Ball, 2005). See All Chapters |
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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 We distinguish between prioritized standards and supporting standards. We must focus our content and curriculum, collaboratively determining which standards are |
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## Chapter 2 Designing Conceptual Units in Mathematics |
Chris Weber | Solution Tree Press | ePub | ||||

This chapter is about 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. See All Chapters |
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## Chapter 5 Integrating Assessment and Intervention |
Chris Weber | Solution Tree Press | ePub | ||||

Success in mathematics is a moral imperative and “algebra is a civil right” (Moses, 2001, p. 5). If educators do not believe in each student’s ability to master mathematics concepts and procedures, then we would have to consider why we are bothering to intervene. We must accept that some students will simply require alternative strategies to learn, that not every student will learn the same way, and that some students will require additional time. We must also believe in our ability to teach every student. A teacher’s sense of self-efficacy significantly predicts the achievement of students, and elementary school teachers’ beliefs in their abilities to teach mathematics lag far behind their beliefs in teaching reading well (Ashton & Webb, 1986; Bandura, 1993; Coladarci, 1992; Dembo & Gibson, 1985). We must also believe, and communicate to students, that mathematics achievement is not dependent on innate ability; work ethic, effort, perseverance, and motivation exert a significant impact on learning (Duckworth, Peterson, Matthews, & Kelly, 2007; Dweck, 2006; Seligman, 1991). See All 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 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. See All Chapters |
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## Appendix Reproducibles |
Chris Weber | Solution Tree Press | ePub | ||||

Teachers and students can use the scoring guide that follows to evaluate responses on an assessment.
Use the following form to diagnose where and why a younger student’s mathematical thinking is breaking down.
Ask the student to orally state the name of a number.
Ask the student to legibly write a number.
Ask the student to represent a number with a set of objects.
Ask the student to orally state or legibly write the number that represents a set of objects.
Ask the student to identify which set of objects represents the greater amount.
Ask the student to count on and count back, beginning at different values.
Ask the student to identify or write the missing value in a sequence. See All Chapters |