The Math Trailblazers approach to all mathematics topics promotes the coordinated development of both procedural skill and conceptual understanding. This is particularly apparent in Math Trailblazers’ approach to computation.
Meaning, Invention, Efficiency, Proficiency
Students will learn and understand when to apply an operation and how to use varied computational methods to solve problems. With each operation, standard methods for solving problems are not introduced until students develop good conceptual and procedural understandings. Research has shown that introducing such procedures too early may short-circuit students’ common sense, encouraging mechanical and uncritical behavior.
• Develop meaning for the operation
• Invent procedures for solving problems
• Choose appropriate and efficient strategies to carry out procedures
• Develop proficiency
The goal of the first stage is to help students understand the meaning of the operation. Most of the work involves solving problems, writing or telling “stories” that involve operations, and sharing solution strategies. These methods typically involve a great deal of mental arithmetic and creative thinking. The use of manipulatives, pictures, and counting is encouraged at this stage. Discussing these informal methods helps students’ understanding of the operation.
In the next stage, the focus shifts from developing the concept of the operation to devising and analyzing procedures to carry out the operation. Students invent methods for carrying out the operation, explaining, discussing, and comparing their procedures. Multiple solution strategies are encouraged and parallels between various methods are explored. When students invent their own methods, it helps make mathematics meaningful by connecting school math methods to their own ways of thinking.
In the third stage, a standard algorithm is introduced. Algorithms are not presented as the official way or the only way to solve a problem but rather as yet another procedure to examine. The algorithms used in Math Trailblazers are not all identical to the traditional ones taught in school. Often these algorithms are easier to learn than the traditional methods, and more transparent, revealing what is actually happening .
Finally, students master efficient and reliable computational algorithms. This procedural fluency is based on a solid conceptual understanding so that it can be applied flexibly to solve problems. These procedures become part of the students’ base of prior knowledge on which they can build more advanced conceptual and procedural understandings.
Practicing the Operations
Students practice using the operations in activities, games, and labs. More practice is provided in the Daily Practice and Problems in K–5, the Home Practice in Grades 1–5, tiered lesson Workshops, and in targeted practice opportunities. See Figure 1 to see the development of the whole-number operations in Grades K–5.
Figure 1: Development of the Whole-Number Operations in Grades K–5
Flexible Strategies for Whole-Number Computation
Even after analyzing and practicing standard methods, students are still encouraged to solve problems in more than one way. Flexible thinking and mathematical power are the goals, not fluency with a handful of standard algorithms. There are many other kinds of computation than the standard paper-and-pencil algorithms for adding, subtracting, multiplying, and dividing, and students need to be able to solve problems more than one way.
There are several rationales for developing flexible strategies for whole number computation in the Math Trailblazers classroom. First, a wide range of strategies allows for sense-making and development of conceptual understanding. Further, when students have a range of strategies to choose from, they develop computational fluency that is more accurate, more efficient, and more flexible. Students are more able to engage in problem solving rather than resorting to using a rote procedure to get an answer. The third rationale for developing a wide range of strategies in the classroom is that when students have access to a variety of strategies, they are better equipped to respond to a problem. A range of strategies helps students to access and respond to different mathematical contexts. For example, they can ask themselves: What type of computation does the problem call for? Is it enough to estimate or do I need an exact answer? Can I use mental math to figure this out? Do I need paper and pencil? Finally, a range of strategies supports a range of student identities and needs. When students have access to a range of strategies, teachers can better differentiate to meet the individual needs of students. Teachers are also able to use information about a student’s use of strategies to better assess his or her mathematical understanding.
Math Trailblazers prepares students to compute accurately, flexible, and appropriately in all situations.