8 Mathematical Practices
- Make sense of problems and persevere in solving them. The two key ideas here are problem solving and perseverance. Children must understand that math problems are doable, but that sometimes they require struggle. As a teacher, you must give students problems that are difficult enough to challenge them to think, and then you have to be willing to stand back and let them grapple. Use questioning strategies to provide support without giving solutions away. As students tackle tough problems, they come to understand that problem solving requires constant checks on both the process and reasonableness of their solution.
- Reason abstractly and quantitatively. Simply put, this means that kiddos need to be able to translate back and forth from symbols and numbers (abstract) to images and words (quantitative). This is accomplished by overlapping the concrete, representational, and abstract stages of learning. Initially, students will model math with manipulatives (concrete) and draw a picture (representational) to show their thinking. In time, they should be able to express their concrete and representational work with numbers and symbols (abstract). It’s important, however, that students also get practice creating stories and pictures to describe abstract concepts. For example, given the equation 2 + 3 = 5, students should be able to model the equation with manipulatives, draw a picture, or write a story problem.
- Construct viable arguments and critique the reasoning of others. Developing this practice requires shifting the emphasis from the solution to the process. The mathematics classroom should be an environment filled with rich discourse, and students should understand without question that they must justify and defend their work with words, objects, and pictures. Be sure to model what respectful disagreement looks like and how to ask probing questions to clarify thoughts.
- Model with mathematics. Proficient mathematicians understand the usefulness of math in their everyday lives. They realize that the purpose of learning mathematics is to solve problems. They use a variety of different models to represent their math, including symbols, pictures, graphs, tables, diagrams, words, and objects, to name a few.
- Use appropriate tools strategically. Students should be introduced to a wide variety of mathematical tools and should become proficient in their use. It’s important, however, that students are able to freely choose the tools they want to use for each job. Too often we, as teachers, choose for our students. Math manipulatives and tools should be available and accessible in the classroom at all times. Just as students justify their processes, they should be able to explain their choice of tool.
- Attend to precision. Mathematicians talk the language of math. They use formal vocabulary, and they understand and can communicate the meaning of symbols, including the equal sign. A proficient mathematician computes accurately, but also efficiently. For example, repeated addition could be used to accurately compute 56 x 8, but that would not be efficient. Mathematicians add detail to their work through the proper use of labels and units of measure.
- Look for and make use of structure. Patterns help students make meaning of math. For example, a student who understands how to compose and decompose 5 will be able to use that information to solve 25 + 5? = 1, or 5 . A student who notices the patterns on a hundreds chart has the foundational understanding needed to build place value skills. Understanding that 3 + 2 has the same value as 2 + 3 sets the stage for fact fluency
- Look for and express regularity in repeated reasoning. The patterns in mathematics help us to be more efficient in our computations. A student who does not understand the patterns in a hundreds chart will likely count each square to add 10. Children learn that they can group objects and skip count, rather than counting one by one. Subitizing the 5 red beads on a rekenrek promotes a count on strategy, rather than a count all strategy. And many mental math strategies rely on seeing the patterns in numbers, such as understanding that 36 + 19 is just one less than 36 + 20.