How often do we ask students to show their mathematical thinking or explain an answer using words, pictures, or diagrams? If you’re like most of us, the answer to that question is probably “Very often”!
But what does it mean to express mathematical ideas and processes through these modalities? What is our expectation that students’ work contains language that connects to diagrams and pictures? Are representations of thinking created after a solution is found, or are pictures, diagrams, and language part of developing a solution? To explore these questions, let’s first take a close look at four mathematical tasks, each requiring increasingly complex conceptual understandings, and the visual representations and language that can support understanding these tasks.