It is important to realize that in order for students to make connections to mathematical concepts and to retain the skills that they are taught from year to year, they must have some meaningful purpose. Mathematical Practice 4 addresses this need by having students model real world situations. The numerical quantities that students are representing should be related to concrete objects in their lives and interpreted in a concrete, pictorial, or symbolic way.

These videos and articles provide examples of mathematical situations where representation is essential to understanding. In these resources students are shown choosing their own solution methods and their own representations. In some cases a variety of representation options are offered to students as guidelines but it is ultimately up the student to show their understanding in a chosen manner. Allowing students to choose their own method of representation is essential to understanding Mathematical Practice 4; by allowing students to model problems in their own pictorial, conceptual, and concrete ways there will be more discussion about these representations, to clarify that the student understands the math concept and has drawn the appropriate conclusion. Two specific articles in this collection address this concern: “Engaging Reluctant Problem Solvers” and “It’s Not Just Notation: Valuing Student’s Representations”.

In order to gain mastery of any topic, students must first understand all of the components involved. In math, this means the scaffolding progression between concrete models, pictorial models, and symbolic models must be incorporated into topics throughout the K-12 education process. It is unlikely that all three representations will be utilized in every unit, especially with younger students who may rely heavily on manipulatives and pictures as their representations. Practice Standard 4 requires all students to model their understanding in a manner that they are comfortable with by representing the problem and solution. Younger students may choose to draw pictures or use manipulatives to represent the problem and to explain their solution, while older students will likely be writing number sentences, equations, or paragraphs to explain their thinking.