Mathematical information and concepts are normally represented in three different forms: symbolic, numeric, and visual. When I do mathematics, I tend to write what I am doing or thinking about in one or more of these forms. I write equations, list tables of values, and sketch diagrams. When I am doing mathematics alone, I think about what these symbols, numbers, and pictures represent. When I teach mathematics, I talk and ask questions about what these symbols, numbers, and pictures represent.
Although the visual component of mathematics has always been important in geometry and trigonometry, the use of graphing calculators and computer graphics software increased the need for visualization in algebra courses too. Many students have difficulty visualizing in mathematics, so helping students to improve their visual thinking skills is one of my major goals as a mathematics teacher.
I assign and grade a lot of student work in any course I teach. I try to select problems and ask questions in a way which requires symbolic, numeric, and visual thinking. I find that many students try to imitate what they have seen in written form when they solve problems. Some students can produce equations, graphs, and diagrams that look like what they have seen, and yet they can neither interpret what their work means nor identify their mistakes. Others do not finish problems because they "cannot remember how it is supposed to look." In short, students tend to focus upon what is written and miss much of what is spoken. As a result, students often think that learning mathematics means learning to write what they see the teacher write - not learning to think mathematically.