The process of analyzing curriculum materials before an adoption is a daunting undertaking. Diane Briars, President of the National Council of Teachers of Mathematics wrote “Curriculum Materials Matter: Evaluating the Evaluation Process” to offer her recommendations for effective curriculum materials evaluation. The article that follows analyzes *Math Trailblazers *4th Edition against her criteria. She also contributed to the development of the Mathematics Content Alignment Tools developed by the CCSS Mathematics Curriculum Materials Analysis Project. To aid in using this tool, we have responded to the questions in Tool 3 and supplied references in the Chapter/Page column of Tools 1 and 2 to lessons in the curriculum where each standard is explicitly linked to an assessment task. These references are by no means a comprehensive list of everywhere the standard occurs; rather, they are snapshots of where each is predominantly featured. Additional tasks and practice corresponding to every standard can be found throughout the curriculum. We urge you to look at all the student materials (*Student Guides* and *Student Activity Books*) as well as the digital *Teacher Guides*, which include all the Daily Practice and Problems (DPP) and Home Practice (HP) and are a rich source of distributive practice.

Tool 1 Grades K–2 | Tool 1 Grades 3–5 | ||||

Tool 2 Grade K | Tool 2 Grade 1 | Tool 2 Grade 2 | Tool 2 Grade 3 | Tool 2 Grade 4 | Tool 2 Grade 5 |

Tool 3 Grade K–5 |

*Math Trailblazers* 4^{th} Edition reflects the goals and ideas of the National Council of Teachers of Mathematics (NCTM) *Principles and Standards for School Mathematics* and provides teachers the support and tools needed to develop students’ proficiency with the expectations outlined in the *Common Core Standards for Mathematics*.

*Math Trailblazers* maintains a careful balance between developing students’ conceptual understanding, procedural skills, and fluency with rigorous problem solving. Based on meaningful, real-world, problem-centered exercises, the program is designed to create an educational experience resulting in students who think flexibly about mathematics, see connections between the mathematics they learn in school and their everyday life experiences, enjoy mathematics, and have the critical-thinking and problem-solving skills required for future success.

Key shifts called for by the Common Core are a greater focus on fewer topics, better coherence that links topics and thinking

across grades, and more rigor, requiring that teachers pursue conceptual understanding, procedural skills and fluency, and application with equal intensity. With this and current research in mind, *Math Trailblazers* underwent a thorough revision in both content and structure to reflect the findings of our research and to accommodate the Standards. The greatest content changes occurred in the development of whole number operations, geometry skills, and fraction concepts. Structural changes make *Math Trailblazers* 4^{th} Edition easier for teachers, students, and families to implement, use, and understand. The result is a curriculum designed to help students develop deep conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they learn to solve problems both inside and outside of the classroom.

## The following are important criteria to consider when reviewing *Math Trailblazers* Fourth Edition:

### The development of *Math Trailblazers* content reflects what is known about how students learn most effectively.

*Math Trailblazers* is a mathematics curriculum created by the Teaching Integrated Math and Science (TIMS) Project. The TIMS Project is at the forefront of current national efforts to improve the teaching and learning of mathematics and science in elementary schools. MTB4 is a research-based program informed by a multi-year field test funded by the National Science Foundation, an Implementation and Whole-Number Study, recommendations by the National Research Council, and more current findings by other researchers like those at the Rational Number Project and the Cognitively Guided Instruction professional development program. The program is also based on research by experts at the Learning Sciences Research Institute (LSRI) of the University of Illinois at Chicago. LSRI’s interdisciplinary researchers work to identify and solve critical learning and instructional challenges in literacy, mathematics, science, and the social sciences, and design and test innovative models, tools, and technologies to support teachers and students. Learn more about the research, development, and revision of *Math Trailblazers*.

*Math Trailblazers* effectively supports students’ learning of the Standards.

The fourth edition of *Math Trailblazers* keeps students’ learning in the forefront by providing instructional support for highly effective teaching and learning. Informed by more than 25 years of research about student learning, the inquiry-based program provides students with the time and opportunities to study mathematics in a powerful way. *Math Trailblazers* helps teachers create a positive and interesting classroom environment and is written to capture students’ attention so they can become fully engaged in their mathematics learning.

One of the goals of good instruction is for students to reveal what they know and what they need. MTB4 builds on students’ own conceptions and ideas, and allows teachers to observe students’ learning, understanding, and misconceptions. Students invent, revise, and compare strategies, solve problems in multiple ways, and decide what makes sense. Research has shown that this leads to better recall, creates more flexible thinkers, and supports higher-level thinking. (National Research Council, 2001)

Lessons are outlined using the learning progression rather than the choreography of the lesson. *Math Trailblazers* thoughtfully considers what type of scaffolding students really need, and spends more time and attention there and less on following a step-by-step, one-size-fits-all procedure. The curriculum engages students with relevant, thought-provoking questions that stimulate interest and evoke mathematical thinking. Interesting, active lessons consistently include problems and meaningful tasks with multiple entry points, discussions in which students compare and revise strategies and solutions, and regular opportunities to make connections with representations.

At the same time, the program guides teachers in what to look for as students work and provides suggestions to support and challenge students. MTB4 incorporates explicit support and instruction with mental math strategies, rationale for learning multiple strategies, the development of flexible and strategic thinking, and the development of math facts and whole number operations.

Students are regularly offered practice connected directly to the content of the lesson, distributive practice (where practice is broken up into a number of short sessions and offered over a longer period of time), and targeted practice (activities that provide a different kind of focused practice to help students meet their goals).

*Math Trailblazers* engaging student materials include clear tutorials, extensive glossaries, and helpful reference sections designed to support student learning. Strategies and representations are presented with clear and explanatory visuals in helpful ways that are not busy and distracting to students. *Math Trailblazers* 4^{th} Edition offers interactive digital complements to the program’s core components that include animated diagrams and examples, step-by-step problem-solving tutorials, and enhanced Adventure Stories.

*Math Trailblazers* materials support teachers’ implementation of effective teaching practices.

**“Improvements in student learning can come only from a strategy focused on improving instruction.” (Noguera 2004; Schmoker)- “Will CCSSM Matter in Ten Years” by Matthew Larson September 2012 Teaching Children Mathematics, NCTM**

A review of MTB4 teacher materials shows that the program attends to teachers as learners in order to help improve instruction. Transforming teaching practices to the new expectations can be challenging. MTB4 helps teachers make these transitions by incorporating clear guidance to support the teaching of the Standards, helpful point-of-use information, a highly effective assessment structure, and rich mathematical background.

Built-in teaching supports such as Content Notes provide information about mathematical concepts that may be unfamiliar to teachers. Suggested prompts help to guide instruction and move learning forward, and possible student responses offer examples of how students typically express their ideas and answers. Lesson plans routinely include strategies to help teachers think about multiple ways to solve problems and conversation prompts to lead discussions about the efficiency of strategies.

Sample Dialogs address misconceptions that students may have during a lesson, and Meeting Individual Needs boxes offer ways to support and challenge learners. TIMS Tips offer logistical suggestions. Summary discussions, clear visuals, and complete answer keys aid in supporting successful implementation.

Each unit is organized around a set of Expectations that are aligned to the Common Core Standards for Mathematics*.* There are numerous opportunities for both formal and informal assessment of student learning, including self-assessment tasks, so that the feedback provided can effectively guide instruction and improve learning. This balanced mix of assessment tasks develops a clear picture of students’ knowledge of the standards-based concepts and skills.

Studies have shown that assessment is most valid closest to instruction. *Math Trailblazers’* formative embedded assessment system is designed to give teachers better information about where they are going, what students know, and how to make instructional decisions to meet students’ needs. While summative assessments are also included in this balanced system, there are extensive opportunities to assess students’ progress at these crucial times, to address needs and adjust instruction.

Helpful assessment tools allow teachers to assess measurable benchmarks written from the Standards. Ongoing Assessment Boxes within each lesson cue teachers to assessment opportunities and explicitly describe what is expected of students. Well-defined tasks designed to elicit direct, observable evidence of the degree to which a student can independently demonstrate the targeted Standard are presented within Check-In Questions, or other types of assessments. Useful tools for recording, monitoring, and assessing student progress are included. To assist teachers interpret student work and efficiently provide frequent and meaningful feedback on their students’ progress, easy-to-use Feedback Boxes are included as well as complete and detailed answer keys. Targeted practice suggestions and differentiated practice such as Workshops are built right into the lessons.

Designed to enrich and transform instructional practices,* Math Trailblazers* is like having professional development at a teacher’s fingertips. The information in the Mathematics in this Unit (or Mathematics in this Month for kindergarten) provides teachers with more detailed practical information about the mathematics content, the design rationale, and the summaries of supporting research that apply in each unit. Assessment in this Unit (or Assessment in this Month for kindergarten) sections describe the assessment structure, offer guidance for observing and assessing students’ work, and summarize targeted practice opportunities in the unit.

*Math Trailblazers* maintains a careful balance between developing students’ conceptual understanding, procedural skills, and fluency.

*Math Trailblazers* is a coherent curriculum that provides students with robust opportunities to develop both the necessary skills and conceptual understandings needed to become mathematically proficient. The importance of developing a strong conceptual foundation is emphasized in research that guided the development of *Math Trailblazers. *There is a careful, deliberate, and progressive development of ideas within each grade. Major topics are linked between grades, and major content is developed over time.

*Math Trailblazers* is based on the idea that mathematics is best learned solving problems in real-world contexts that are meaningful to children. Students need proficiency with the basic skills in order to do so. Both concepts and procedures are important, and neglecting one will undermine the other. The balance between these two areas is important because students who do not develop conceptual understanding are more likely to rely on procedures and may be less likely to engage in robust problem solving and reasoning about mathematical ideas. In order to acquire the skills, *Math Trailblazers* students build on their own knowledge. Then they use informal, invented strategies to solve problems. Finally, students are able to revise their strategies and develop fluency with formal procedures.

The iceberg diagram illustrates the importance of effort spent on the conceptual development of mathematical concepts. Students spend time inventing and using pre-formal and informal strategies as they stimulate prior knowledge, make connections, deepen their understanding, and develop the “floating capacity” necessary for them to use and apply formal notations and strategies.

MTB4 pays explicit attention to the conceptual development of mathematical concepts and gives students time to make meaning of the mathematics before asking them to practice procedures. If students are asked to grapple with, represent, and justify their own thinking and reasoning, they develop a deeper understanding of the concepts and become better problem-solvers. *Math Trailblazers* students work actively and collaboratively with their peers to make sense of concepts, while using multiple representations and tools to solve complex problems. They repeatedly make connections among these strategies and representations to deepen their mathematical understanding. Students have to be able to make connections among representations, and then choose the appropriate tool and strategy to be able to solve problems efficiently. Students in a *Math Trailblazers* classroom are given regular opportunities to explore, develop, discuss, and refine their strategies and mathematical ideas. This solid conceptual foundation gives students the power to develop fluency with mathematical procedures and the ability to think flexibly and strategically about mathematics.

In order to develop a deep conceptual understanding, a high degree of procedural skill and fluency, and the practical application of mathematics skills to solve problems, *Math Trailblazers*4^{th} Edition focuses on addition and subtraction concepts, skills, and problem solving in Grades K–2. In Grades 3–5, the focus is on concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions. Students are routinely provided with opportunities to apply concepts and skills, and regularly presented with modeling, reasoning, and problem-solving experiences to develop their fluency with procedures. Students’ learning is connected across grade levels to create a coherent curriculum in which students continually build mathematical understanding.

*Math Trailblazers* approaches math facts in a conceptual and systematic way, and provides explicit support and instruction with the strategies that develop math facts and whole number operations. Students first approach basic facts as problems to be solved using and developing strategies. Students then practice these facts in small groups organized by strategy. As students become proficient with these strategies, they become fluent with the math facts.

**“Practicing single-digit calculations is essential for developing fluency with them. This practice can occur in many different contexts, including solving word problems. Drill alone does not develop mastery of single-digit combinations. Practice that follows substantial initial experiences that support understanding and emphasize ‘thinking strategies’ has been shown to improve student achievement with single-digit calculations”(National Research Council, 2001).**

*Math Trailblazers* 4^{th} Edition offers regular opportunities and time for practice and includes both practice connected directly to the content of the lesson and distributive practice within each unit and at all grade levels. Opportunities for practice include tiered lessons, individualized practice and challenge in Workshops in Grades 1–5, Home Practice in Grades 1–5, Work Stations in kindergarten, and Daily Practice and Problems and Homework in all grade levels. Studies have shown that “premature practice can be detrimental but that properly managed practice is essential in the development of expertise. . . . Brief, engaging, and purposeful practice distributed over time is usually most effective” (Isaacs & Carroll, 1999).

*Math Trailblazers’* deliberate progression in conceptual development, purposeful use of representation, and explicit instruction about making connections among representations exists to help teachers analyze developing mathematicians. Opportunities for real-world modeling, problem solving, and reasoning, the routine application of concepts and skills, and regular practice of procedures enables students to become mathematically proficient.

### MTB4 students have opportunities to regularly engage in problem solving, reasoning, and modeling, and to practice the “habits of mind” related to both problem solving and communication.

Mathematical content is connected to mathematical practices in MTB4 by incorporating lessons that include meaningful, challenging tasks or problems.

The lessons and tasks that *Math Trailblazers* students are involved in:

**Build mathematical proficiency****Challenge and enrich student thinking****Develop mathematical vocabulary related to the ideas and concepts****Promote the ability to learn multiple ways of thinking about and representing mathematical ideas (pictures, diagrams, concrete models, numeric/symbolic)**

**- NCSM Great Tasks for Mathematics K–5 (Schrock, Norris, Pugalee, Seitz, & Hollingshead, 2013)**

MTB4 lessons are comprised of rich problems worth solving that involve both mathematical understanding and procedural skill. These high-quality tasks engage students in strategic problem solving, reasoning, modeling, and making sense of mathematics as core instructional activities rather than special features of a lesson. They are integrally interwoven throughout the lessons, in every grade, throughout the entire curriculum, and are not problems scattered here and there, or tacked onto a set of exercises as an afterthought. These interesting problems permit multiple entry points and approaches to address a variety of learners, allow multiple solution strategies, and effectively address the learning goals of the lessons.

The content in MTB4 is the result of the application of the habits of mind in the CCSSM Mathematical Practices. The six Math Practices Expectations in *Math Trailblazers* embody the Standards for Mathematical Practices in the CCSSM. These Expectations remain constant throughout the curriculum and define those “habits of mind” that are related to both problem solving and communication. The content is taught through these practices.

*Math Trailblazers** *Math Practices Expectations are written using “student friendly’ language so that students can routinely use them to help them focus on practices related to both problem solving and communication. Students become more comfortable with these Expectations by reviewing them before, during, and after problem solving and by deciding how well they are met in their work and the work of others.

While the Standards for Mathematical Practice and opportunities for students to model, reason, and problem solve have always been an integral part of the *Math Trailblazers* curriculum, opportunities for their application are now more explicit and meaningful. The Standards for Mathematical Practice are identified and well connected to the content being addressed. Extensive instructional support for teachers and students such as prompts to guide instruction, assessment tools, and Math Practices reference materials help students effectively engage in these practices.

A unique component of *Math Trailblazers* is the use of a laboratory method that helps students build and analyze mathematical models of authentic situations, providing opportunities to reason quantitatively, construct viable arguments, attend to precision, and persevere as problem solvers. Students use multiple representations (e.g., objects, graphs, tables, pictures, and equations) to describe and analyze the mathematical situation, allowing them to access the mathematics, apply conceptual understanding, and develop problem solving strategies. The contexts for solving problems help the students in *Math Trailblazers*^{®} classrooms model real world situations and learn to use and apply mathematics in meaningful ways.

### The focused materials and coherent content of MTB4 allow students to develop a deep understanding of mathematical concepts.

CCSSM requirements were carefully considered during the revision of *Math Trailblazers**.* MTB4 lessons and units target the major work of each grade. Any other material is brief and supports or connects to the major work of the grade. The content builds on previous understandings and provides opportunities for students to make important connections. The rigorous curriculum and challenging mathematics presented balances regular opportunities for students to apply mathematical concepts in real-world situations, apply models, and choose appropriate solution strategies. Students’ conceptual understanding is developed as they use multiple representations, work on rich tasks, and write and speak about their understanding, and then students practice performing mathematical procedures quickly and accurately.

Learners need opportunities to explore, develop, build upon, revise, solidify, and make their own meaning of a topic. This requires time. *Math Trailblazers* is not designed to “cover” a topic in one unit or one month because that is not how students learn. MTB4 is organized into focused, coherent units with narrow sets of Expectations and content. Units are organized around what is assessable, and content and skills that are interrelated. With fewer major topics per grade level, students can explore a topic, create their own meaning, revisit the topic throughout the year, build on their understanding, and continually advance in their learning. Students’ learning is connected across grade levels to create a coherent curriculum in which students continually build mathematical understanding. The content is organized so that students can clearly see how ideas build upon and connect with other ideas both within and across grade levels.

Substantial revision was made to the curriculum by a team of mathematicians, scientists, education researchers, teachers, and teacher developers. The team carefully considered the scaffolding students needed, which topics required more time and attention, and whether content ought to be omitted. The small team worked closely together, which enabled extremely collaborative and coherent work across lessons, units, and grade levels. The result is a curriculum in which students concentrate on the major topics identified in the standards for each grade and are given the time and support to develop the identified proficiencies.

In order to develop deep conceptual understanding, a high degree of procedural skill and fluency, and the practical application of mathematics skills to solve problems, the content in Grades 1–5 is organized into focused units with a narrow definition of Expectations. Grades K–2 focus on addition and subtraction concepts, skills, and problem solving. In Grades 3–5, the focus is on concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions. Each unit consists of several lessons. A lesson is defined as a progression that leads to an assessment point. The kindergarten content is a little different in that it is organized by month, and within each month there are 3–6 lessons. Each lesson contains a progression of activities. Some activities are designed to be repeated to provide multiple opportunities for students to develop conceptual understanding. Students in all grades are routinely provided with opportunities to apply concepts and skills, and regularly presented with modeling, reasoning, and problem-solving experiences to develop their fluency with procedures.

For more detailed information on MTB4 and the CCSSM, see MTB4’s Focus on Critical Areas and Focus on Major Work.

*Math Trailblazers* considers equity, diversity, and access, and has high-quality content and instructional practices to guarantee the success of all learners.

A *Math Trailblazers* classroom includes high-quality and interesting mathematics instruction for all learners based on the belief that all students deserve a rich and challenging math curriculum. High expectations and strong support for students and teachers allows a wide range of learners to access the rigorous mathematics in *Math Trailblazers*. MTB4 includes new and more explicit strategies and tools to help teachers and students succeed. One such tool is the point-of-use Meeting Individual Needs boxes within the lessons that offer ways of dealing with misconceptions and varying student needs, including challenges for students who are ready to advance. A variety of instructional approaches are used in each lesson including using multiple representations, a range of questions, models, sample student work, frequent embedded checks for understanding, flexible grouping, targeted practice, and Workshops to reach a vast range of learners.

The curriculum allows students to develop deep insights into important concepts instead of experiencing a more cursory treatment of a broader range of topics. It avoids instruction that forces over-reliance of memorization and instead emphasizes conceptual understanding. The lessons are intentionally designed and organized to clearly focus on the content, the process, and the products so students develop understanding of the key concepts, principles, skills, and facts of math. This intentional attention to conceptual development of mathematical concepts and to the sequence and organization of the curriculum supports all students as they learn.

The authentic tasks within *Math Trailblazers* lessons are challenging, and involve problem solving and reasoning that is appropriate for all learners. Helping students access math in varied ways, building on prior math knowledge, making connections across math topics, moving from concrete to representational to abstract, using multiple representations, providing many examples, offering manipulatives, attempting new strategies, and communicating solution tasks can increase accessibility for some students, and motivate and challenge all learners.

*Math Trailblazers* 4^{th} Edition attends to increasing diversity within the classroom by incorporating most of these features daily in lessons:

- Engaging student materials include clear tutorials, extensive glossaries, and helpful reference sections designed to support student learning. Strategies and representations are presented with clear and explanatory visuals in helpful ways that are not distracting to students. Interactive digital components have animated coaches, diagrams, and examples, step-by-step problem-solving tutorials, and enhanced Adventure Stories with “read-to-me” options.

- Hands-on, concept-based instruction serves to provide access and motivation to all students. An engaging, problem-solving focus allows students to find alternative strategies and approaches that work best for them. Students are able to access problems differently, and use a variety of strategies for doing computation. The introduction of more than one valid procedure for each arithmetic operation offers students alternative options that align better with their learning strengths. Challenging reasoning and problem-solving tasks are solved using a range of strategies that allows more advanced thinkers to use more sophisticated strategies.

- Rich problems add meaningful context that helps students make sense of the mathematics, build upon their prior knowledge, and judge the reasonableness of their answers.

- Authentic tasks develop mathematical abstract reasoning, develop metacognitive skills, and increase motivation and interest in mathematics as students do the work of real mathematicians.

- Extensive and ongoing use of multiple representations of arithmetic operations such as manipulatives, pictures, models, and other visual representations support students by providing different entry points for understanding the operations. Students are explicitly asked to use tools and representations. They learn when and how to use them to discover what’s most efficient for them and which tools and representations work for different problems.

- Communication of mathematical ideas using different representations helps students understand and retain concepts when they rephrase problems in their own words, communicate their solution strategies, and summarize what others have reported. Connections to literature, problems that involve everyday life, and opportunities to read, write, and communicate about problems provide meaningful access to rigorous mathematics.

- A conceptual and systematic approach to the instruction of the math facts focuses on patterns and connections among and between facts rather than rote memorization for better retention. Students first use and develop strategies to solve the facts, and then regularly practice these facts in small groups organized by strategy with Daily Practice and Problems in Grades K–5 and Home Practice in Grades 1–5. As students become proficient with these strategies, they become fluent with the math facts.

- Workshops, games, and labs incorporate a variety of strategies to address varying needs and provide another source of motivation or practice that can be used effectively with all students. Differentiated practice questions and variations of games help address a variety of students’ needs.

- More opportunities and time for practice within each unit include both practice connected directly to the content of the lesson and distributive practice in the form of Daily Practice and Problems in Kindergarten through 5
^{th}grade and Home Practice in Grades 1–5. Some DPP problems are TIMS Challenges.

- Carefully selected homework tasks with instructions to families, Letters Home, and access to online tools, reference sections, and glossaries provide support to students and families.

- A balance of whole class instruction, varied small group work, and individual work combined with a supportive environment promotes understanding through discourse with props, collaborative work, and communication.

*Math Trailblazers* 4^{th} Edition includes new and more explicit strategies and tools to help teachers meet the needs of the diverse learners in their classrooms. There has been substantial revision of the whole number strand in all grade levels, incorporating explicit instruction with mental math strategies and better support and rationale given for multiple strategies. There is more attention given to the development of flexible and strategic thinking, and helpful Strategy Menus in Grades 1–5 for students, teachers, and families to aid in this process. Attention is routinely drawn to these tools that can assist learners to develop the needed conceptual understanding of the operations in order to gain fluency with the procedures.

Some activities are specifically designed to provide students with focused intervention, or targeted practice, rather than simply more of the same practice. In Workshops, independent practice is individualized to reteach, practice, or challenge individual students. Work Stations for kindergarteners offer additional opportunities to gain independence and practice skills and concepts worked explored as a class. These targeted and differentiated practice activities are included in every unit.

To help guide instructional decisions, embedded assessment opportunities are regular, consistent, and clearly indicated so individualization can be easily implemented to support each child in his or her mathematical development.

Well-defined assessment tasks are presented within quizzes, tests, and Self-Check and Check-In Questions. In Ongoing Assessment Boxes, assessments are described with learning performance descriptors called Expectations. Targeted Practice recommendations are included here, too, to ensure all students grasp important concepts. Feedback Boxes are included with many assessments to provide timely and constructive feedback. Assessment in this Unit documentation describes the assessment structure, includes useful tools for recording, monitoring, and assessing student progress, and summarizes targeted practice opportunities in the unit.

*Math Trailblazers*^{ }gives teachers the power to be intentional and make choices.

With MTB4, teachers are encouraged to be intentional, to listen to their students, and to make choices. There is additional practice if students need it, targeted practice, and differentiation options. The materials allow teachers to make instructional decisions that will meet the needs of their students and provide practice that helps advance all learners.

Because standards vary across the states, there may be content in MTB4 beyond what is addressed in some standards. This extra content does not have to disrupt the focus or coherence of the materials. Upon revision, it was determined that this content can help students learn what is in the standards and be enriching. Any units targeting additional and supporting concepts are brief to ensure that students will spend the strong majority of the year on the major work of the grade. For example, 3^{rd} grade students use the context of volume to solve multiplication and division problems with larger numbers. A context such as volume does not detract from the focus of multiplication and division. Rather, this secondary story supports the main topic while giving students an interesting, meaningful context for real-world problems. With this design, students have the opportunity to access material such as volume or area through natural contexts that are developmentally appropriate without it distracting from the main topics of the grade level.

Ultimately, the authors of *Math Trailblazers* know that good instruction depends on knowledgeable and flexible teachers who plan lessons carefully. Teachers have the power to use MTB4 in the way that best meets the needs of their students, and to make choices about omitting content if they feel that is beneficial. The authors of MTB4 believe that it is easier to eliminate additional content than it is to design rich material in meaningful contexts to fill in gaps in a curriculum.

### MTB4 provides a balanced portrayal of various demographic and personal characteristics, and includes globally diverse, high-interest stories and settings.

Classrooms are growing more diverse. Students have familial ties to other nations, speak different languages, carry different strengths and challenges, and have unique cultural traditions and family structures.

*Math Trailblazers* materials promote equity by portraying a diverse population of students in several fictional classrooms. Colorful and stimulating pages are filled with students of various demographics, personal characteristics, and special strengths. *Math Trailblazers* students learn and grow with these characters. The characters share their unique experiences, struggles, strengths, insights, strategies, and solutions throughout the curriculum. Many of these characters are introduced in kindergarten. They reappear and grow as students grow throughout their elementary grades. These characters are not special. Rather they are all simply important members of an elementary classroom, with different opinions and perspectives to contribute, just as in real life.

To ensure access to all, *Math Trailblazers* problems are carefully written to include contexts that appeals to all regardless of background — school functions, everyday life, parties, sports, animals, and games. Adventure Stories, however, often involve exciting contexts that include a variety of geographical regions and global cultures. Contributions from diverse societies, cultures, and civilizations are highlighted in these illustrated stories to introduce mathematical concepts. For example, students are introduced to ancient Chinese number puzzles called Magic Squares to develop problem-solving skills and addition strategies in *Yu the Great: A Chinese Legend*. In *Phil and Howard’s Excellent Egyptian Adventure*, two boys learn about an ancient Egyptian multiplication strategy. Students review representations and partitioning of numbers, place value, regrouping, and multidigit addition and subtraction strategies as they learn about the life and discoveries of the famous Italian mathematician, Leonardo Fibonacci, in the story *Leonardo the Traveler*. In *Peanut Soup*, George Washington Carver and his students plan a luncheon where they use fractions to convert recipes to the needed size. In the story *A Matter of Survival*, students are transported to the rain forest and introduced to population sampling. Later in fifth grade, students use connections between ratios, proportions, and graphing as they read *Bats*, set at Bandelier National Monument in New Mexico.

*Math Trailblazers* succeeds in engaging and appealing to a multitude of students by incorporating interesting and varied contexts. The curriculum has always placed importance on providing a balanced portrayal of various demographic and personal characteristics.

** **RESOURCES

- Isaacs, A.C., and W.M. Carroll. (1999) “Strategies for Basic-Facts Instruction.”
*Teaching Children Mathematics*, 5 (9), 508–515. - Larson, Matthew. (2012) “Will CCSSM Matter in Ten Years.”
*Teaching Children Mathematics*, NCTM. - National Council of Teachers of Mathematics. (2000)
*Principles and Standards for School Mathematics**.* - National Research Council. (2001) “Developing Proficiency with Whole Numbers.” In
*Adding It Up: Helping Children Learn Mathematics.*Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academies Press. - Pellegrino, J.W., and S.R. Goldman. (2007). Beyond rhetoric: Realities and complexities of integrating assessment into teaching and learning. In
*The future of assessment: Shaping teaching and learning*. C. Dwyer (Ed.). Mahwah, NJ: Erlbaum. - Schrock, C. Norris, K. Pugalee, D., Seitz, R. and F. Hollingshead. (2013)
*NCSM Great Tasks for Mathematics K–5*. - Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding.
*Mathematics Teaching in the Middle School,*14(2), 110–113.

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