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Math Trailblazers BSD Math Project Team Questions As a leading publisher of research-based mathematics programs, Kendall Hunt is proud of the successful results students all across the country have achieved through the use of our integrated, inquiry-rich, standards-based programs. Math Trailblazers provides your students with the tools and educational skills necessary to achieve and st remain competitive in the 21 Century global economy. Our Math Trailblazers program embodies the National Council of Teachers of Mathematics Principals and Standards for School Mathematics and is based on the ideas and research that mathematics is best learned by solving problems in real-world contexts and that a curriculum should balance conceptual understanding and procedural skills which lead to computational fluency. (Kendall Hunt has attempted to answer BSD questions by grouping them together by topic.) 1. How is Trailblazers challenging? Math Trailblazers introduces challenging content within a coherent progression of mathematics at each grade level and across grades: computation, whole numbers, fractions, measurement, data collection, statistics, geometry, ratio, probability, graphing, algebraic concepts, estimation, mental arithmetic, and patterns and relationships. Contexts for this demanding content begin with students’ lives. Students are actively involved, grappling with significant concepts while learning and applying basic skills and procedures. Lessons are grounded in everyday situations, so abstractions build on experience. By presenting mathematics in rich contexts, the curriculum helps students make connections among real situations, words, pictures, data, graphs, and symbols. The curriculum also validates students’ current understandings while new understandings develop. Students can solve problems in ways they understand while being encouraged to connect those ways to more abstract and powerful methods. This allows students to apply the skills and concepts in an authentic way which builds a deep understanding of mathematics. As an example of how Math Trailblazers presents challenging content in a clear and thoughtful progression which spans Kindergarten through Fifth Grade, we can look at the teaching of fractions and decimals, which are critical foundations for algebra. Concepts are taught by using a careful developmental approach to ensure and promote the coordinated development of both procedural skills and conceptual understanding. Students in grades K-3 establish links between symbols and referents for part-whole fractions. Early on students are making connections between different representations to promote a solid foundation of the mathematical understanding of fractions. Decimals are also treated as an alternative notation for certain fractions at the primary level. In Grade 3, students continue to develop concepts of the unit, but they begin to use more varied models. Translating between common fractions th and decimals is stressed. This prepares the 4 grade student for operations with fractions and decimals. Students solve problems involving addition and subtraction with the aid of manipulatives and other fractional models to establish a deep understanding and a basis for algorithmic work. In Grade 5, paper- and-pencil procedures for addition, subtraction, and multiplication of fractions and decimals are explored, including use of common denominators and reducing. Many contexts are used to reinforce student’s understanding of fraction and decimal concepts in challenging yet real world contexts, giving students multiple opportunities to apply their understanding. For example in Grade 5 in the lesson “Comparing Lives of Animals and Soap Bubbles”, students collect 1 data about the life spans of soap bubbles. As they analyze the data, they convert the data first to a fractional quantity, then to the equivalent decimal fraction. Making connections between mathematical concepts and real-world situations helps students make sense of the mathematics they are learning. Mathematical expectations in Math Trailblazers are high and grow steadily. Review of concepts and skills is carefully built into new and increasingly challenging problems as the program builds upon itself both within and across grade levels. Every content strand within Math Trailblazers interweaves the promotion of conceptual understanding with distributed practice of skills and procedures. 2. You emphasize “real” problems, where can I find examples of what you think “real” problems are? 3. Problems? Math Trailblazers challenges students to solve complex “real” problems. These problems come in a number of different forms, including multidisciplinary, open-ended problems, in many kinds of situation that students may encounter daily. Math Trailblazers expects students at all grade levels to identify problems, think through solutions and alternatives, and explore new strategies. Examples of ”real” problems are those in which students have to reflect upon what is being asked and design effective strategies in response to the question. They are posed in an engaging context. They challenge students to think in new ways. They are authentic. They are learning opportunities to go from the concrete to the abstract. Real world problems are naturally interdisciplinary. In Math Trailblazers the real-life problems integrate topics that are traditionally separated. Interesting problems permeate the entire program, providing a context that promotes conceptual understanding, computational fluency and problem solving skills. In Grade 3 Unit 7 “Exploring Multiplication and Division”, students encounter many different types of problems and learn to communicate their solutions in various ways. For example students use graphs to explore multiplication and division in Lesson 1 Lemonade Stand and number lines in Lesson 4 Birthday Party and in Lesson 5 The Money Jar students work with interpreting remainders with division. And finally in Lesson 6 Walking around Shapes students integrate multiplication, division and graphing in an investigation between the perimeter and length of one side of regular polygons. Fractions are also introduced. For example students are asked to share six pizzas equally among 12 people. They find that each person will receive ½ of a pizza. The resulting number sentence shows students in a real world context that fractions denote division. Experiences with many types of problems will provide a strong conceptual foundation not only for multiplication and division but also for fractions, ratios, and proportional reasoning. 2 4. Assessments? Math Trailblazers Assessment If one word might be used to sum up the assessment program in Math Trailblazers it would be, “Balance.” The only fair way for a teacher to determine what a student knows is to incorporate a diverse, balanced array of assessments techniques into the classroom everyday. In Math Trailblazers there are paper and pencil assessments, performance assessments, and observational assessments. Formative and summative opportunities abound in every grade. Example: To get an in-depth picture of student progress and achievement, a variety of assessments are available in each unit. For example, in Grade 4 Unit 8, a student can be assessed by: Facts Assessments using Facts Inventory Tests in the Daily Practice and Problems Midyear Exam that focuses on skills and concepts learned in the first half of the school year Assessment Blackline Masters that focus on specific skills and concepts Observational Assessment using the Observational Assessment Record Groupwork Assessments during labs and games using the Assessment section in the Teacher Implementation Guide Open-response and constructed response problems to be scored using the Solving, Knowing, and Telling rubrics Targeted Questions in Activities/Labs/Games to be scored using the TIMS Multidimensional Rubric Collection Folders and Portfolios to show growth over time For a full summation of the assessment opportunities in Math Trailblazers, see the Assessment section of the Teacher Implementation Guide. 3 5. How exactly does this curriculum/program integrate language arts and science? Language Arts Language arts are incorporated into the program in a number of ways. Many lessons engage students through the use of trade books or through mathematical adventures and historical stories found in Adventure Books. The Adventure Books are available as audio CDs to support a variety of learners and is designed in an engaging comic book style. Most labs, activities, Adventure Book stories, and games ask students to share their thinking orally or in written form. Open-ended response questions are the rule, rather than the exception. Students use rich language as they explore mathematics, becoming stronger communicators as they discuss strategies, practice mathematical terms and record solutions. It is the belief of the developers of MTB that students need to read, write, and talk about mathematics, all elements of mathematical discourse including mathematical terms. See the Language in the Math Trailblazers classroom section of the Teacher Implementation Guide for a full discussion of how Math Trailblazers utilizes discourse in the classroom. Science Beginning in kindergarten, data collection and analysis form an important piece of mathematical content. In first grade and continuing through fifth grade, students use an adapted version of the scientific method, the TIMS Lab Method, as a structure for learning. It is the process of science that is incorporated into the program, as an avenue for work with meaningful, authentic numbers. While most of the labs are physical science in nature, some are related to other science disciplines. Each grade of Math Trailblazers contains 7-9 applications of the scientific process. 6. What are the ways that science/language arts are integrated with the math programs? How does it affect students? Much of this is answered earlier. As for the effect on students, it helps make the study of mathematics real and applicable to student lives. Educators have observed that students understand why they are learning mathematics and do not question the value of what they are learning. An engaged student is an active learner. The science and language arts components provide this connections piece. 7. I’d like to know about science integration and materials. The TIMS Laboratory Method is integrated in lessons throughout each grade. This scientific method includes: “Scientific Notebooking” is used to record the lab process (including manipulated, responding, and fixed variables) Collect the data in tables Graphing the data in bar point graphs Exploring the data in tables and graphs, looking for patterns and relationships Discussing and analyzing the results, drawing conclusions, making extensions, and connections 4 Regarding materials, a full manipulative kit is available for purchase through Kendall Hunt Publishing. This kit includes needed scientific materials such as graduated cylinders, balances, gram masses, eyedroppers, etc. 5 8. Background information? Background information/Teacher Support Math Trailblazers offers background information and teacher support in a number of different ways. Background The Background section in each Unit Resource Guide (URG) provides in-depth information about the mathematical content in the unit. Elsewhere in the URG, Content Note boxes are seen throughout the lessons, providing background tips at the point of use. In kindergarten, the Teacher Resource Book contains background information for every content strand. The Teacher Implementation Guide, in all grades, provides a series of TIMS Tutors, an explanation of mathematical concepts designed to support teachers in their facilitation of learning. Each URG contains a Unit Planner, a powerful tool to help teachers prepare for each lesson within the unit. Within many Math Trailblazers lessons, there is a section called the Curriculum Sequence. This section tells you what students have learned in the past (and when and where they learned it) as well as what they will learn in the future (and where and when they will learn it.) For example, Grade 2 Unit 2 Lesson 5 explores the subtraction facts. The Curriculum Sequence tells you that students developed subtraction facts strategies in Units 8, 13, and 20 in Grade 1 and will continue to study the facts in the Daily Practice and Problems in every unit throughout the remainder of Grade 2. This background information can assist you in planning. Another tool available to help teachers quickly find related background information is the curriculum wide index which covers the content in both the student and teacher materials. Teacher Support Kendall Hunt is dedicated to working with you to design and implement a long-range support program that fits your needs. We also offer a number of different areas of ongoing support, including the Math Trailblazers ListServ, the Math Trailblazers Voice online newsletter, and www.mymathtrailblazers.com 9. How is it seen? Math Trailblazers is a balanced program. It provides a flexible and balanced instruction that is student-centered, teacher-directed, with both collaborative and individual work. It is based on the student’s needs supported by powerful tools for teachers to support all learners. The intent of Math Trailblazers beginning in Kindergarten is that all students be prepared for secondary algebra, geometry, problem solving and reasoning and proof. Students must solve problems, but of course they need basic skills to do so. Both concepts and procedures are important and neglecting one will undermine the other. There is a place in the curriculum both for paper-and-pencil algorithms, mental arithmetic and estimation. This approach is supported by the National Mathematics Advisory Panel’s FINAL REPORT p. xix: “To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem solving skills.” 6 The national math advisory panel has suggested specific grade-level bench marks for a critical foundation for algebra. These benchmarks are well aligned with the development and mastery sequence of math facts and whole number operations within MTB. Students not only achieve these benchmarks but retain the skills through deep understanding of concepts and on going practice. 7 10. The program is correlated to national standards. How about correlation to Oregon’s newly adopted K-8 standards? We have completed the required correlations to the Oregon standards and the information is available on the BSD Web site. Kendall/Hunt is also currently developing a reverse alignment to the new Oregon standards. This correlation will align standards to every unit and every lesson at every grade level K-5. It will be a valuable tool for teachers to identify how the standards align to the following for each unit. o Lesson o Daily Practice & Problems o Assessments o Games We will then make recommendations on any enhancements or adjustments needed. This reverse alignment will be completed by January 2009. The work is being done by an educator who is very knowledgeable of the Oregon state standards and Math Trailblazers. 11. Claim 20 years of research. What was the scope and depth of the research? The research began over 20 years ago when the National Science Foundation (NSF) awarded grants to the Teaching Integrated Mathematics and Science (TIMS) Project at the University of Illinois in Chicago. These grants were designed to assess the efficacy of the TIMS labs in the areas of mathematics and science, throughout Chicago and surrounding communities. The results were so impressive that the NSF, a decade or so later, awarded the TIMS Project one of the three elementary school grants to create what has become Math Trailblazers. Research has continued over years, including the ARC study of the effectiveness of standards-based mathematics. Research continues to this day, analyzing the results of districts around the country using the program. Each new Math Trailblazers edition has been revised with teacher feedback, current mathematics research, and the changing math standards. See the Foundations of Math Trailblazers section in the Teacher Implementation Guide for a thorough explanation of Math Trailblazers research history. 12. Materials provided in multi-languages? Kendall/Hunt offers Math Trailblazers in English and Spanish. All student components are available in both languages as well as all blackline masters from the teacher materials that would be used with students. The Letters Home in all of the Unit Resource Guides are provided in both English and Spanish on the same page. Kendall Hunt will work with any users who would like to translate Math Trailblazers into other languages. You may contact your sales representative with these requests. See the Teaching the Math Trailblazers Student: Meeting Individual Needs section of the Teacher Implementation Guide for more information on teaching students in multi-language classrooms. 8 13. ELL support other than bolded words? See the Teaching the Math Trailblazers Student: Meeting Individual Needs section, especially Part I (Using Math Trailblazers with English Language Learners), of the Teacher Implementation Guide for more information on this subject. There are SIOP and Sheltered teaching strategies in most lessons like strong use of manipulatives, graphic organizers, cooperative work and clear learning objectives with new vocabulary stated at the beginning of every lesson. These features make the lesson easily adapted to ELLs. Question #15 addresses each of these strategies in more depth. 14. Language dependent vs. language development? Math Trailblazers uses real-life contexts to help students understand math concepts and promote student interest and engagement in the mathematics. However, incorporating problem solving contexts into the lessons requires increased use of language through speaking, listening, reading and writing. Language dependent contexts provide an opportunity for students to make sense of symbolic representations; e.g.; 9x4 becomes 9 groups of 4. However, using language in mathematics results in language development thus improved literacy. Students will only learn the language of mathematics if there is significant use of productive (speaking and writing) and receptive (listening and reading) forms of language. This is also true for the ELL student. Everyday language is improved while using it within the context of math class. Math Trailblazers provides teachers with reading comprehension strategies, support for non or reluctant readers. The Language in the Math Trailblazers Classroom (Section 12) in the Teacher Implementation Guide serves as a comprehensive instructional tool for teachers. It cites research, offers multiple strategies for supporting and developing language and gives valuable examples of how teachers provide access for all students. 15. Could you discuss integrated differentiation in more detail? With problem solving central to the design of Math Trailblazers, opportunities to differentiate instruction are built into the program in multiple ways. Lessons are designed with the flexibility to reach students with a wide range of abilities and learning styles. Some opportunities to differentiate instruction are specifically identified, other opportunities are integrated throughout the curriculum, including: Science The hands-on nature of scientific investigation allows children with different learning styles and interests to explore the mathematics. Language Arts Math concepts are introduced and developed using contexts rich in language. Children can explain their thinking and solutions to problems in a variety of ways, using both oral and written communication. Manipulatives Using manipulatives provides different representations of math concepts to help children better understand and remember the mathematics. Collaborative Work Students with different abilities and styles learn from one another and gain confidence in their own problem solving abilities. Balanced Assessment 9 Multiple forms of assessment provide students with a variety of ways to show progress to their teacher. Curriculum Sequence Concepts are revisited in different contexts within the grade and across all grades, giving students repeated learning opportunities. 16. How easy is the differentiated instruction for the teacher to actually do? How much work is it for them to prepare a differentiated lesson? Besides the multiple-entry points with each Math Trailblazers lesson and the differentiation opportunities described in earlier questions, there is also a planning tool to help classroom teachers and classroom support staff implement differentiated instruction. Each unit in Grades 1–5 includes an overview of the differentiation opportunities in the unit. This overview is called “Teaching All Math Trailblazers Students.” TaG, SpED and ELL students are taken into consideration along with many other unidentified groups of students needing differentiation. It is the belief of Math Trailblazers that teachers have a lot on their plates without asking them to create entirely different lessons for students in their classrooms. If a problem is a good one, it reaches the needs of a range of learners by providing multiple entry points as well as options for multiple solution strategies. This does not mean that every student does exactly the same thing. Many of the lessons are already “tiered” for use in a heterogeneous classroom. This enables teachers to make informed instructional decisions based in their knowledge of their students and on the essential mathematics of the lesson. Also built into lessons are challenge problems, additional practice problems, and suggestions for extending the lessons. To better help teachers understand how a differentiated lesson is realized in a Math Trailblazers classroom, see the Teaching the Math Trailblazers Student: Meeting Individual Needs section of the Teacher Implementation Guide. To assist teachers in meeting the variety of student needs in their classrooms, the Differentiation Resource Book, based on the Math Trailblazers curriculum, was developed to help deepen students’ mathematical understanding. The new Differentiation Resource Books for Grades K-5 (available November 2008) are an addition to the lesson extension opportunities already available in Math Trailblazers. Within this handbook, a chart lists the “existing” opportunities for extending the lessons, such as Journal Prompts, Literature or Software Connections, and “Existing Extensions” in the program, as well as new “Additional Extensions” compiled in this book. These extensions are further categorized to assist you in selecting those activities that best meet the needs of your students. Activities that review skills and concepts already learned are annotated with a backward arrow. Activities that support your current lessons and mathematical explorations are marked with an up arrow. And finally, activities that expand upon your students’ knowledge and challenge them to think in new ways are marked with a forward arrow. Pages can be duplicated as blackline masters for the students and the Answer Keys are on separate pages for the teacher. 17. How can students work ahead if they are advanced? Rather than thinking of working ahead, think of working deeper into the mathematical content. The Daily Practice and Problems’ Challenges do just that. They are in the context of the lesson, yet carry the 10 exploration to a deeper level. The “Extension” section, contained in many lessons, continues this. The Differentiation Resource Book contains further ideas. Examples: For an example of a Challenge DPP, see DPP item H in Grade 4 Unit 8. This DPP asks students to find the area of a rectangle when given the width and the perimeter of the rectangle. This is more challenging than the problem on the Midyear Test in Grade 4 Unit 8 that asks students to find the area and perimeter of a shape drawn to scale with square-inch tiles. For an example of an Extension, see Grade 2 Unit 2 Lesson 1 “Tile Designs.” The lesson explores tile designs where when given a design, students write a number sentence. The Extension challenges students to work in reverse. Given a number sentence, they must generate a matching tile design. Then, explain why it is that more than one tile design can be generated to fit the number sentence. One example in the Differentiation Resource Book is Kindergarten Month 2 Lesson 1 where students identify patterns. The activity in the Differentiation Resource Book challenges students with more complex patterns including symbolic patterns. 18. What online resources/games are available for the students? Within many of the Math Trailblazers lessons, there are Connections. These Connections often contain suggestions for utilizing software that aligns with the current math topic or links to websites that offer additional support. The Teacher Implementation Guide contains a compiled list of these suggested Connections for online resources/ games for students. Also, mymathtrailblazers.com carries links to additional sites. www.mymathtrailblazers.com contains the following resources for students: Student books (Student Guide, Discovery Assignment Book and Adventure Book) December 2008: Audio version of the Adventure Books (currently available on CD) Online Homework Help for families and students Virtual Manipulatives (December 2008) (other links suggested in Connections currently) Student Games (working on development of these for first quarter 2009) For teachers, www.mymathtrailblazers also contains: Geometer’s Sketchpad Activities aligned to each grade level (Grades 2-5) that teachers can use alongside Math Trailblazers or as extensions (free software download available on the site for MTB users) Professional Development tips, videos, and information on trainings offered, the ListServ and other resources for teachers Curriculum Maps which summarize how the skills and concepts build and grow within and across grade levels 11 Mymathtrailblazers.com provides a new level of online teacher, administrator, family and student support. For administrators, the online leadership handbook provides powerful tools to help the administrator or math leader meet the challenges of successful implementation. For teachers, mymathtrailblazers.com gives them instant access to professional development. This digital resource provides instant, convenient, on-line professional development-anytime, anywhere. 12 Many resources are grade-level specific. Meet varied student needs with online differentiation resources. Access lessons and activities for children who need to revisit concepts. Include a wealth of engaging extensions for further challenges. Lesson planning is just a click away with the online Unit Resource Guide. Identify lesson subjects, supplies needed, and time required with the Unit Planner. Linked to blackline masters and synchronized to the online student edition. For families, the online Family section supports the successful partnership between school and home. The online Homework Help section provides background information, suggested approaches, and examples for families. For students, the online grade specific resources for the Math Trailblazers student includes textbooks in English and Spanish as well as other digital resources to enhance student learning. Find the digital resources you need anytime, anywhere at mymathtrailblazers.com! Math Trailblazers is an award-winning program that combines 20 years of research and dynamic digital learning resources for you and your students. 13