2013 NCTM Conference (4/18/13 - 9:30 am - Marc Garneau):
The second session I attended at the 2013 NCTM Conference was Marc Garneau's "I See It: The Power of Visualization". As a visual learner, I know how helpful it can be to provide students with more concrete representation of mathematical concepts.
Marc suggested that most students have been taught symbolically their whole lives, and the lose touch with what the symbols really represent. Students who do not understand what the symbols really mean have no idea how to visualize math or even to apply it in real life applications. Visualizing math can help students reason about math and give them a fuller understanding of the abstract concepts.
A typical example of visualization is creating diagrams to depict fractions. These can be very helpful building students' understanding of fractions, which can be a very challenging concept for students. Comparing fractions can be difficult symbolically, but is much simpler if students are able to depict fractions visually and comparing the concrete representations of these numbers.
A more exciting example of visualization for high school teachers is the use of patterns to develop algebraic reasoning. Teachers can use visual patterns to first help students develop recursive reasoning (i.e. "look at these two first figures in a pattern: how many squares will the third figure have?") , and then to build to a more abstract understanding of algebraic patterns. Teachers can use patterns to prod students to reason about what the 10th or 100th item in the pattern will look like, without recursively counting the items in each step. This helps them understand algebraic modeling without even realizing what they are doing.
These patterns can be very simple or quite complex. Marc showed us a quadratic pattern that can be used to help students begin to explore quadratic functions, which could be a much more tangible introduction to this family of functions.