Differentiating Math Instruction, K–8: Common Core Mathematics in the 21st Century Classroom

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William N. Bender

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    Acknowledgments

    Corwin gratefully acknowledges the contributions of the following reviewers:

    • Marcia Carlson
    • Sixth-Grade Teacher
    • Crestview Elementary School
    • Clive, Iowa
    • Julie Duford
    • Fifth-Grade Math Teacher
    • Polson Middle School
    • Polson, Montana
    • Esther Eacho
    • Associate Faculty
    • Johns Hopkins University
    • Baltimore, Maryland
    • Debi Gartland
    • Professor of Special Education
    • Towson University, Department of Special Education
    • Towson, Maryland
    • Edward C. Nolan
    • Mathematics Supervisor, PreK to Grade 12
    • Montgomery Public Schools
    • Rockville, Maryland
    • Rachel Spenner
    • Sixth-Grade Teacher
    • Westridge Elementary
    • West Des Moines, Iowa

    About the Author

    Dr. William Bender is an international leader in instructional tactics, with broad expertise in areas including project-based learning, technology in the classroom, differentiating instruction, response to intervention (RTI), as well as other areas dealing with general classroom instruction. In particular, Dr. Bender has written more books on response to intervention than any other author in the world, and two of these are best-selling books on that topic, and one was a 2010 finalist for the Distinguished Achievement Award for Excellence in Educational Publishing. By the summer of 2012, Dr. Bender had completed eight books and a professional development videotape on various aspects of RTI. He also served as a consultant to hundreds of districts and many states as they established their RTI plans. In the fall of 2010 he was selected to work with the Ministry of Education in Bermuda to establish their nation-wide RTI framework.

    In addition to book development, Dr. Bender consistently receives positive reviews of his professional development workshops for educators at every level. He has an innate ability to use a combination of practical strategies and easy humor, to keep educators informed and engaged, eager to hear more. Dr. Bender's books and workshops provide research-proven, practical strategies, and convey this information in a humorous, motivating fashion.

    Dr. Bender began his education career teaching in a junior high school resource classroom, working with adolescents with behavioral disorders and learning disabilities. He earned his PhD in special education from the University of North Carolina. As a professor of Education, he has taught in higher education around the nation including Bluefield State College, Rutgers University, and the University of Georgia. He is now consulting full time, writing several new professional development books. Dr. Bender has written over 60 research articles and 26 books in education, including his most recent books:

    • Cool Tech Tools for Lower Tech Teachers: 20 Tactics for Every Classroom
    • Project-Based Learning: Differentiating Instruction for the 21st Century
    • The Teaching Revolution: RTI, Technology, and Differentiation Transform Teaching for the 21st Century
    • Differentiating Instruction for Students With Learning Disabilities: New Best Practices for General and Special Educators (3rd ed.)
    • Response to Intervention in Math

    Educators are invited to communicate directly with Dr. Bender if they wish at ewilliamb@teachersworkshop.com. Educators may also follow Dr. Bender on Twitter (@williambender1), when he posts exclusively on educationally related content, his books or educational workshops, notices of other professional development (particularly free PD) opportunities, and other educational topics.

  • Appendix A: Recently Developed or Widely Used Curricula in Mathematics

    There are a plethora of innovative instructional, computer-based software programs and recently developed hard-copy mathematics curricula used in schools today. Many of these, like the Khan Academy, have been described in text, and some are free for teachers to use. This appendix presents several recently developed, broad-scale mathematics curricula that many mathematics teachers are currently using. Many of these tools allow for individual laptop or tablet-based instruction, and might well provide the option of flipping the classroom, as discussed in Chapter 3. Others have proven to be a great foundation for Tier 2 and 3 RTI interventions in mathematics as discussed in Chapter 7. While other curricula could have been included here, these were chosen for the reasons above, or because they are frequently used in schools today.

    Number Worlds

    Griffin (2003a, 2003b, 2004a, 2005) has provided another research-based structure that represents children's early understanding of numbers, Number Worlds. This research-supported resource for teaching young students mathematical concepts is a mathematics readiness/math curriculum that teaches foundational mathematics concepts and skills on a conceptual level and an application level (Griffin, 2003b, 2004a, 2005). This curriculum is intended for children from prekindergarten through Grade 1, and includes software, manipulatives, problem-solving scenarios, games, lesson plans, and hard-copy workbooks, all of which focus on developing a hands-on understanding of numbers and how numbers are used in the real world (Griffin, 2004b; Griffin et al., 2003). While predating the Common Core State Standards for Mathematics, this curriculum is nevertheless congruent with those standards, in that students are taught mathematics readiness and number sense in a manner that stresses deep conceptual understanding. Number Worlds is now published by Scholastic Research Associates (http://www.sranumberworlds.com).

    SRA Number Worlds is based on five instructional principles. The program was developed to

    Build on the student's current knowledge using multilevel activities;

    Use natural learning paths that children typically employ to develop number sense;

    Present new knowledge in a way that supports the common progression children use when learning numbers;

    Teach computational fluency; and

    Stress hands-on exploration, problem solving, and communication using numbers.

    Each concept in the curriculum is discussed and utilized in various ways to assure flexibility in students' understandings of math constructs (Griffin, 2003b, 2004b). In addition to being recommended for average achievers, this program has been evaluated with children from low-income populations, as well as children with special needs, and has proven effective in enhancing number sense, computational fluency, and reasoning, as well as performance on traditional, standardized mathematics achievement tests (Griffin, 2004a, 2005). Positive results of this program were evident in some of these studies, up to one year after termination of the program (Griffin, 2004a, 2005).

    Differentiated activities are relatively easy to develop using Number Worlds. Teachers can adjust the sequence of work to accommodate students with many different learning styles. Teachers will assess each student's level of ability and select specific activities for the individual children throughout the year (Griffin, 2004a, 2005).

    SAS Curriculum Pathways

    One recently developed online curriculum for older students that is being used increasingly in schools is the SAS Curriculum Pathways (http://www.sascurriculumpathways.com). This curriculum is free for teachers worldwide and is used by more than 50,000 teachers in over 12,000 schools. This program provides an extensive, online curricula with materials and instructional activities in English, language arts, science, social studies, mathematics, and Spanish. The lesson activities are designed around the Common Core State Standards in math and are appropriate from grade levels 6–12. While the curriculum is free, teachers are required to login to access the material, and a brief tutorial video is available free of charge at the website above. The company behind this website designed the curriculum materials extremely carefully, with content experts in various subject areas consulting on all aspects of the curriculum. A wide variety of instructional activities are incorporated into this curriculum, as well as excellent graphics and short video demos that can be used in flipped classes. Every mathematics teacher in the middle grades should explore this free curriculum.

    Math in Focus: Singapore Math

    This comprehensive math curriculum, by Marshall Cavendish, is based on instructional practices developed for mathematics in the nation of Singapore in recent decades (http://www.hmheducation.com/singaporemath/index.php). This curriculum has been developed for students from kindergarten through Grade 8 using a concrete, pictorial, and abstract framework to cover mathematics concepts in depth. In the Math in Focus curriculum, beginning in kindergarten, students are exposed to mathematical concepts, covering less content overall but at a greater depth, with a strong emphasis on conceptual learning as well as mathematical procedures. Rather than covering a concept in two or three days, students in this curriculum may spend two to three weeks exploring mathematical content much more fully, and students achieve mastery prior to moving to the next big idea. Because of this emphasis on deep understanding, this curriculum was one of 15 mathematics curricula that were reviewed in the development of the Common Core State Standards in mathematics. For that reason, this curriculum has and will continue to receive increasing attention.

    Accelerated math for Intervention

    This newly developed commercially available curriculum from Renaissance Learning utilizes the widely popular Accelerated Math curriculum that has been around for a while, but adds an overlay of assessment and progress-monitoring tools to make this very appropriate for use in RTI interventions (http://www.renlearn.com/am/RTI.aspx). Accelerated Math for Interventions enables monitored, differentiated math practice, and is recognized as a mastery measure by the National Center on Response to Intervention. Reports from this curriculum provide daily information about student progress toward math mastery (TOPS report), skill by skill, and a variety of other reports (e.g., Status of the Class Report) are available to allow teachers to see student performance or compare students' performance. Because Accelerated Math has been widely used for years, and with the growing emphasis on specific targeted interventions in mathematics for students with mathematics deficits, teachers should expect to see this curriculum more frequently across the elementary grade levels.

    IXL

    IXL is a subscription-based site that presents mathematics exercises for grades kindergarten through middle school (http://www.ixl.com). Students completed activities online, answering simple questions on various topics, and their progress is tracked online. IXL assesses students' understanding as they practice, and then generates detailed performance reports that can be used for instruction and/or progress monitoring in an RTI framework. The various reports include data on grade-level proficiency, trouble spots, and even progress toward mathematics mastery.

    VMATH

    Vmath was developed by the Voyagers company, a company known for the Voyagers reading program (http://www.voyagerlearning.com/curriculum/math-solutions/vmath). Note: Be careful to not confuse Vmath with Vedic Math!). Vmath is a supplemental curriculum and benchmarking tool, supported by a number of case study applications (found at the website above) that is aimed at students who are struggling in math from Grade 2 through 8. Vmath presents a balance of teacher-led, explicit instruction, printed materials, assessment options, and student work online and is intended to fill gaps in student knowledge across the grade levels. In the daily lessons, teachers model the skills, facilitate both group and individual practice, and provide corrective feedback as students experience difficulty. Each lesson addresses conceptual understanding, skills practice, and problem solving. The program is founded on mathematics standard from the National Council of Teachers of Mathematics and uses CBM as one of the assessment tools. The program aligns well with RTI requirements, and can be used either in Tier 1 instruction or in Tiers 2 and 3 for specific targeted interventions. Also, intensive training and support are provided once a school undertakes implementation of this program.

    Study Island

    Study Island, from Archipelago Learning, is a supplemental web-based instructional program offering instructional practice in almost every subject area, including math. This is tied specifically to each state's standards of learning (making this particularly useful for states that have not adopted the Common Core State Standards) as well as the Common Core. Study Island is also tied to each state's testing program (visit http://www.studyisland.com for more information). By basing this curriculum exclusively online, the Study Island developers have made these instructional and assessment materials available to licensed users, including teachers and students, both at school and at home, and this can be a significant advantage. Some students will undertake these instructional activities in the home environment, and software programs that are loaded exclusively on computers at school do not have that option.

    Instructional activities are flexible, and teachers can either allow students to select topics on which to work, or they can assign specific topics that are based on the exact learning needs and styles of the individual student. The student will then be presented with various computer-based work or educational gaming activities.

    Based on adaptive assessment technology within the program, the curriculum adjusts itself according to the learning curve of particular students, either moving students through the reading content faster or moving students into a slower track with more practice on various content items. Once students have mastered a particular lesson and assessment, they receive a blue ribbon and are able to move to the next lesson. However, should a student receive a low score, the program may prompt the student to continue working on the same skills until he or she develops proficiency. Thus, students receive instructional feedback each time they answer a question. This can be implemented as either a stand-alone instructional program or as a supplemental program in conjunction with other instruction. Thus, this program can also be used as the basis for RTI procedures either as Tier 1 instruction or as a Tier 2 or Tier 3 intervention in mathematics across grade levels. The stand-alone nature makes this curriculum intervention and assessment program ideal for RTI implementation in elementary, middle, or even secondary schools. Like more modern computerized programs, reports can be generated either for individual students or the entire class.

    Study Island has only limited anecdotal research support, and that research was prepared by an independent firm under contract with the company. The supporting research can be found on the Study Island website (http://www.studyisland.com). The Study Island website does present a variety of reports suggesting how Study Island can be implemented in the context of RTI, which will help schools considering RTI implementation in the future. A number of schools around the nation are using Study Island, with some success, in the RTI context.

    Transmath

    TransMath (http://www.voyagerlearning.com/cs/Satellite/transmath) is a higher-level, hard-copy supplementary mathematics curriculum developed by John Woodward and Mary Stroh that focuses on moving students from elementary mathematics skill levels up to algebra readiness. This curriculum is intended for students functioning at or below the 40th percentile in Grades 5 through 9, but the content covers a wider range of skills, reaching from number sense to algebraic expression. This curriculum covers fewer overall topics than most core mathematics curricula but covers those topics in much more conceptual depth, making it consistent in intent with the Common Core Standards in Mathematics. This program progresses in three levels that focus on specific mathematics areas: number sense, rational numbers, and algebraic expressions.

    Three placement assessments (one for each of the levels mentioned) come with the curriculum, and each instructional unit within each level also includes two performance assessments. Together, these assessments allow for frequent progress monitoring, making this a useful curriculum in the RTI context. There is limited research support for this curriculum. This curriculum has been implemented by many elementary, middle, and high schools in their RTI efforts, and given the transitional nature of this curriculum, a focus that is virtually unique in the mathematics area, this curriculum is likely to be implemented in many more schools.

    The Successmaker Math Curriculum

    SuccessMaker Math is one component of the broader SuccessMaker curriculum from Pearson Learning. It is an instructional software core curriculum that is available for school or school district purchase. This curriculum provides individualized instruction for elementary and middle schools students in a variety of areas including mathematics (http://www.pearsonschool.com/index.cfm?locator=PSZk99). SuccessMaker is currently being used in many RTI intervention programs in math. Students take an initial assessment when they begin, and those data are used to place each student in specific levels in mathematics. As students complete the lessons, the levels and questions get increasingly complex to move students toward mastery. In addition, SuccessMaker Math generates a variety of progress-monitoring reports for teachers that make it possible to review student growth individually, in subgroups, or for the entire class. This is excellent data for RTI implementation, and the teachers can closely monitor students' growth. This program can be used with the entire class, or students needing a Tier 2 or Tier 3 intervention are able to work with SuccessMaker on an individual basis, at their own pace, with a customized program.

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