Q: Why do students need to have multiple strategies?
A: When students are exposed to multiple strategies, they are better able to find a strategy that works well for them and that suits the problem. A range of strategies allows for sense-making and development of conceptual understanding, promotes computational fluency, helps students access and respond to mathematical contexts, and supports all student identities and needs.
Q: Why do students need time to explore, invent, and discuss? Why can’t they just learn the operation?
A: As students explore concepts with manipulatives, stories, pictures, counting, mental arithmetic, and creative thinking, they begin to understand the meaning of the operation. As students invent methods for carrying out the operation, explain, discuss, and compare their procedures, it helps make mathematics meaningful by connecting school math methods to their own ways of thinking. In time, standard algorithms are introduced and students master efficient and reliable computational algorithms. Students are more ready to understand these algorithms and the mathematical procedures they represent after they have developed a solid conceptual understanding.
Q: Isn’t the best and easiest way the traditional algorithm?
A: Not necessarily. Often the traditional algorithms are more difficult to understand than others and result in many computational mistakes. The algorithms used in Math Trailblazers are not all identical to the traditional ones taught in school for a reason. Often these algorithms are easier to learn than the traditional methods, and more transparent, revealing what is actually happening in the solution. The end goal is not for students to have a single strategy. In order for them to be flexible thinkers, they have to have all these strategies to be proficient. Proficiency isn’t the speed of solving problems; it is thinking flexibly. To learn more, see Whole-Number Computation.
Q: Why are students asked to show their work in math?
A: Studying mathematics is a social activity, and being able to communicate about math is important. Explaining a solution strategy is a vital part of understanding mathematics. Students need to understand others’ ways of thinking so that they can make connections and compare strategies. Recording work helps students recall their steps as they share their strategies with others, especially as problems become more complex. Equally important, teachers need to be able to understand a student’s thinking in order to help them. If a teacher just sees an incorrect answer, it does not show where or how a student went wrong. If a student’s work is shown, a teacher can better see if the error came from miscalculations or if there is a greater misconception about the mathematical content. Math Trailblazers avoids drudgery by not requiring that students show or tell how they solved every problem, nor does the program require students to use a one-size-fits-all reporting system. Students are able to pick how they explain their work in a way that makes sense to them and to the way that they solved the problem. For example, this might be a drawing to show how a student represented the problem on a number line or solved a problem with circle pieces, or it might be in words, written or verbal.
Q: Can’t students learn the facts just by using flashcards and lots of time tests?
A: Math Trailblazers students do use flashcards and take some time tests as they study their math facts, but the program involves much more. Math Trailblazers creates a careful balance between strategies and drill. Practice and assessment of the math facts is based on the results of research in the field. Research has shown that in order to learn the basic facts efficiently, gain fluency with their use, and retain that fluency over time, an approach in which students develop strategies for figuring out the facts rather than relying on rote memorization is required. Students use direct modeling, counting, and reasoning strategies as they acquire operational understanding and fluency with the math facts. This not only leads to more effective learning and better retention, but also to the development of mental math skills. To learn more see Math Facts.
Q: Is using a calculator cheating?
A: No. The calculator is a tool used in appropriate situations to help students explore number ideas and relations, solve more complex problems, and explore mathematics on their own. In Math Trailblazers we assume that the students have access to calculators. Sometimes the calculator is an appropriate tool and sometimes it isn’t. Communicate to students that there are certain things they can do in their heads, there are certain things they can do with pencil and paper, and there are certain things for which they might want to use a calculator. Students need to figure out when to use which one. Prohibiting calculators doesn’t makes any sense as there is no evidence that prohibiting them helps kids learn arithmetic any faster.
Q: Does MTB4 meet the CCSS?
A: Yes. Math Trailblazers 4th Edition aligns with and incorporates all of the Common Core Standards for Mathematical Content and Mathematical Practice while staying true to the Math Trailblazers’® philosophy of providing rich, meaningful contexts for students to develop mathematical understanding, strategies, and number sense. See Meeting the CCSS with MTB4.
Q: How are Math Trailblazers Expectations different from Standards?
A: The Expectations in Math Trailblazers are derived from Standards. The Standards are broad; the Expectations are the measurable, observable, and more specific steps that you need to get the end points, which are the Standards. Expectations help teachers see what it looks like to become proficient with these steps, and show where attention needs to be focused.
Q: What are Benchmark Expectations?
A: Several Expectations in a unit are prioritized as Benchmark Expectations because they are particularly important concepts and skills to be learned in that unit. Benchmark Expectations identify the concepts and skills that are central to the unit, are prerequisites for learning content in an upcoming unit, are the last significant opportunity for assessment in the curriculum or grade, or are concepts commonly associated with difficulties or misconceptions. These Expectations are marked with a red circle in the Key Assessment Opportunities Chart.
Q: Why is there content present in a grade level that is not in the Standards?
A: With MTB4, teachers are encouraged to be intentional, to listen to their students, and to make choices. 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, 3rd 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. It makes sense to introduce some concepts gradually but not as a major topic so that students’ understanding of the idea can grow. Supporting content is introduced at a developmentally appropriate time. 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.
Q: Why are there Adventure Stories in Math Trailblazers?
A: Children make sense of the world through stories. Adventure Stories are interesting, illustrated stories designed to captivate students’ interest and imaginations in a thought-provoking, entertaining medium. They appear throughout the curriculum to introduce and highlight important mathematical concepts in Grades 1–5. Once students are engaged, they care more about the problems waiting to be solved. Adventure Stories make the mathematical content relevant so that students can make connections. The more connections are made, the more students retain the content in their long-term . To learn more, see Math Content in Adventure Stories.
Q: Why are there labs in Math Trailblazers?
A: Math Trailblazers’ repurposed laboratory investigations focus on developing core content for each grade level and help students build and analyze mathematical models of authentic situations. The labs provide opportunities to reason quantitatively, construct viable arguments, attend to precision, and persevere as problem solvers. Students use 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 students learn to use mathematics in meaningful ways.
Q: What if I can’t get through all of the material?
A: The curriculum intentionally includes more material than a teacher can teach in a school year. This allows teachers to listen to their students and to choose the materials that will move them forward. MTB4 has more practice added so that teachers do not have to invent their own, but that does not mean that every problem needs to be completed. Similarly, every problem should not be discussed, reviewed, or graded. Teachers must make instructional choices in order to meet the particular needs of their students. Learn more specific tips and strategies to make instructional pacing decisions for your class.
Q: What is formative embedded assessment?
A: Research has shown that assessment given closest to instruction provides timely and relevant feedback that helps students move forward. Math Trailblazers provides numerous assessment opportunities, both formal and informal, throughout the lessons so that the feedback provided can effectively guide instruction and improve learning. Instead of a system that relies only on summative assessment after student work is complete, this balanced approach gives teachers better information about where they are going, what students know, and how to make instructional decisions before it is too late. With formative embedded assessment, teachers are given time to inform instruction and better meet the needs of their students. Learn more about assessment.
A: There are two reasons. The first is meaning. Developing learners need a concrete example of the abstract concept of number. We call them bits, skinnies, flats, and packs to anchor students back to the concrete. Consider what 10 tens means to a Grade 1 student who is just starting to recognize a unit larger than 1. First, students group real objects (e.g., beans) into groups of ten and leftover ones. To help transition toward the more abstract idea, we replace the real objects with the base ten piece representations. We use a bit for one unit, a skinny for ten units, a flat for one hundred units, and a pack for one thousand units.
Some students need to move back and forth between the terms continually for clarity. If you listen to children, they find the meaning more clearly with the names “bits” and “skinnies” than they do with tens, ones, etc. The goal is to develop meaning and let students use language that makes sense to them to express that meaning.
Secondly, if we use these terms, the representations can have multiple meanings. For example, when students are adding, a bit is one unit, a skinny is ten units, and a flat is one hundred units. However, when students are learning about decimals, a bit can now represent one hundredth, a skinny can be one tenth, a flat can be one whole unit, and a pack can be ten units.