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.
Q: Why do students play games in mathematics class?
A: Playing mathematics games is an engaging way for students to practice skills and strategies. Research has shown that games are an effective and motivating way to deliver this practice. Teachers can easily adapt games to meet the varying needs of their students, and games can be sent home for extra practice. Students also have plenty of in-class practice and homework opportunities.
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.
Q: How does Math Trailblazers address the Common Core Standards for Mathematics?
A: Math Trailblazers team of the Teaching Integrated Math and Science Project at the Learning Sciences Research Institute at the University of Illinois at Chicago and the authors of the Common Core Standards share a common vision— that all teachers and students have access to high-quality learning materials. We created Math Trailblazers Fourth Edition (MTB4) with that shared goal in mind. MTB4 is designed to engage students in challenging, problem-solving contexts that reveal the thinking of developing mathematicians and build on that knowledge to formalize understanding.
Based on research and field test data, our curriculum underwent substantial revision by a team of mathematicians, scientists, education researchers, and teachers. This work did not result in a skeletal curriculum designed to simply cover the Standards. MTB4 was written with the belief that all children deserve a challenging mathematics curriculum and an educational experience resulting in students who enjoy and think flexibly about mathematics, see connections between the math they learn in school and everyday life, and have critical-thinking and problem-solving skills applicable to other disciplines and required for future success. MTB4 proves that a curriculum can support students and teachers to meet the expectations outlined in the Standards while acknowledging students’ desires to engage in relevant and interesting problem solving.