Do The Math: Arithmetic Intervention by Marilyn Burns
According to the 2007 National Assessment of Education Progress (NAEP) Mathematics test, 61 percent of American's fourth graders are not proficient in mathematics. The NAEP data also reveals that 68 percent of eighth graders are not proficient in mathematics. More students in the U.S. need to be proficient in mathematics in order to be successful in algebra. The National Mathematics Advisory Panel's Final Report (2008) establishes fluency with fractions and other basic arithmetic concepts and skills as critical foundations for algebra.
One percent of school-age children experience a math disability not associated with any other learning disability, and two to seven percent experience serious math deficits. Students with mild disabilities do not perform as well as their peers without disabilities on basic operations, and this discrepancy in performance increases with age (Cawley, Parmar, Yan, & Miller, 1996). In addition, students with math disabilities may respond with lower self-esteem, avoidance behaviors, and decreased effort. Learning math is also a challenge for many English language learners, as the content presents its own unique academic vocabulary and is often presented abstractly.
The No Child Left Behind act requires that all students reach proficiency in math by 2014, and the National Council of Teachers of Mathematics (NCTM) goals (2000) aspire for all students to become mathematical problem solvers, learn to communicate and reason mathematically, use representations to model problem situations, and make connections among mathematical ideas. In addition, the National Mathematics Advisory Panel recommends that math curricula for elementary and middle school be a coherent progression of key topics with an emphasis on proficiency. For many students, especially those who struggle, meeting these goals presents a challenge when they only receive the typical 50 minutes a day dedicated to math instruction. Moreover, many students require instruction that is specifically designed to meet them at their level and to focus on the most critical foundational mathematical concepts.
Do The Math addresses these learning challenges facing American students The program's instructional design applies what is known about reaching a wide variety of students who struggle with math to achieve proficiency with arithmetic concepts and skills, by incorporating the following guiding principles: scaffolded content, explicit instruction, multiple strategies, gradual release, student interaction, meaningful practice, assessment and differentiation and vocabulary and language.
Scaffolding is the systematic process of analyzing the content and partitioning it into small manageable chunks for the purpose of planning and delivering instruction that facilitates students' learning.Scaffolding calls for identifying and sequencing the concepts and skills that are essential to the content being taught. Once the content is scaffolded, instruction can then be organized in a way that supports students' learning and paced to allow students sufficient time and practice to be successful. Research shows that scaffolding content to inform instruction benefits all students, and particularly students with learning disabilities.
Do The Math focuses on the basics of Number and Operations with lessons that build accuracy, efficiency, and understanding. All lessons have been carefully designed and sequenced to align with the scaffolding of the content, and then paced to ensure student success.
Using a range of teaching strategies and contexts to teach concepts and skills helps ensure that all students learn and make connections.
Approaching mathematical knowledge through the use of modeling with manipulatives, interacting with mathematical ideas through literature, engaging in discussion of math ideas and skills through games, and viewing and creating visual representations gives the best possible chance for all students to build number sense, develop skills, and deepen their mathematical understanding.
In Do The Math, lessons engage students with concepts and skills in multiple ways using concrete manipulative materials, games that reinforce and provide practice, selected children's literature that provides a context for mathematical concepts and skills, and visual representations to help students represent their thinking.
In Gradual Release pedagogy, the focus of instruction is on the level of responsibility that the teacher maintains to ensure that students understand and can complete a particular task on their own.
The gradual release process begins with modeling new content, followed by guided practice in which students take on increasing cognitive responsibility. This is followed by paired practice, giving students the opportunity to talk to each other about their reasoning to solve a problem, and finally students are released to work independently. This process of moving through phases from dependence to independence has been shown to be an effective strategy, ensuring optimal learning and achievement.
In Do The Math, gradual release pedagogy sets an expectation for student involvement and gives learners the direction and the support needed to be successful. It consists of four phases. In Phase 1, the teacher models and records the appropriate mathematical representation on the board. In Phase 2, the teacher models again but this time elicits responses from the students. During Phase 3, the teacher presents a similar problem. Students work in pairs to solve the problem. The teacher records their solution on the board. Finally, in Phase 4, students work independently, referring to the work recorded on the board if needed.
When students voice their mathematical ideas and explain them to others, they extend and deepen their understanding of the mathematics. Interactions help students make sense of what they are doing and help them to clarify, explain, and evaluate their own thinking and the thinking of their partners.
Pairing students to interact with each other encourages each of them to take responsibility for their own learning as they discuss their thinking when they disagree or do not understand their partner's reasoning. Student interaction can occur in whole groups, small groups, or with pairs of students solving a problem together, playing a game, practicing how to explain how they solved a problem, or reporting to the class how they solved a problem. The opportunity for students to express their math knowledge verbally to a partner is particularly valuable for many students who are developing English language skills.
In Do The Math student interaction is built into the program. One essential routine that encourages active student engagement is think, pair, share. Having students talk in pairs provides them a safe way to share ideas, brainstorm, and practice what they will say when they share with the larger group. Partner interaction is always encouraged when students are released to work independently in their WorkSpace books. Games encourage active engagement and provide practice.
Meaningful practice is practice that is based on conceptual understanding, number sense, and connected to previously learned concepts and skills.
Practice helps strengthen and reinforce what has been taught and learned. Practice that is meaningful is based on number sense and understanding rather than learning and practicing rote procedures. Mathematics makes more sense and is easier to remember when students connect the new knowledge to existing knowledge and solve problems in ways based on understanding of Number and Operations. Meaningful practice provides students opportunities to strengthen and reinforce their learning and maximize their success.
In Do The Math, practice is an essential part of every lesson. The written practice in the WorkSpace is always similar to what students experienced during the lesson. The practice has been carefully sequenced so that no new knowledge or skill is required in order for the student to be successful. Practice is supported through visual directions on the WorkSpace pages for those students who need a point-of-use reminder of the steps involved.
Assessment and Differentiation
Differentiation is an instructional approach based on the principle of equity-that all students, regardless of their personal characteristics, backgrounds, physical challenges, language challenges, and learning challenges, must have opportunities and support to learn.
Differentiation of instruction is essential in order to meet the needs of all students and is a significant challenge for classroom teachers. However, there are some students for whom the differentiation provided during regular math class are not sufficient for them to be successful. Formative assessments are key for identifying these students and their needs. Then, intervention is required to provide these students instructional support in addition to their regular classroom instruction. During intervention instruction, students will progress at different rates and sometimes need additional support. For this reason, an intervention program must also include specific suggestions for differentiating instruction to provide for the success of all students.
In Do The Math lessons are carefully built on scaffolded content with attention to the common misconceptions of students who are in need of intervention. Ongoing assessment and suggestions for differentiation are integral to the program.
Vocabulary and Language
Teaching students correct mathematical language gives them the tools to articulate their mathematical thinking coherently and precisely. Research shows that explicit instruction in mathematics vocabulary supports success with math problem solving.
While many of the words that are used to describe mathematical ideas are familiar to students, their meanings in general usage are often very different from their mathematical meanings. Mathematical vocabulary is determined by social convention, in contrast to the logical foundations of mathematical ideas, which call for thinking and reasoning. This distinction is key to making instructional decisions. Because there is no way to figure out, for example, that numbers divisible by 2 are called even numbers, the best pedagogical choice is to provide that information to the student, that is, to teach by telling.
Explicit vocabulary instruction introduced after students develop conceptual understanding of a mathematical topic, idea, or property makes the vocabulary more meaningful to the student as it connects the new word or words to the mathematics that the student has already experienced.
Students incorporate the new vocabulary into their own language as they explain their thinking to each other or to the whole group when they hear the word or words used consistently and regularly by the teacher and other students.
Explicit vocabulary instruction not only benefits native English speaking students, but is particularly helpful to English language learners. Access to appropriate and effective instruction of academic vocabulary supports English language learners' second language development and facilitates their understanding of the math instruction. Explicitly teaching vocabulary and then using the words frequently in class discussions benefits all learners and encourages them to use the words when they are explaining their reasoning to each other and to the larger group as well.
In Do The Math, vocabulary is introduced after students experience and develop a firm understanding of the mathematical concept so that they can anchor the word in their understanding. The meaning of a key vocabulary word is explicitly taught using the routine of see it, hear it, say it, write it, and read it. The word is recorded on a math vocabulary chart with examples so that students may refer to it as needed. Students read the meaning in their own student glossaries and record the meaning with an example in their WorkSpace books. Students hear the word used frequently by the teacher and naturally begin to incorporate it into their own explanations as they talk to their partners and share their reasoning with the whole group.