Results for: “Mathematics”

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).

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?”

In today’s world, economic access and full citizenship depend crucially on mathematics and science literacy.

—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).

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.

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.