Multiple Representations and Perspectives

There are often multiple ways to think about the same thing. So it can help to be flexible: learn as many different perspectives as possible, and internalize your favorite ones!

For example, you can often use diagrams and pictures to gain intuition: a good portion of our brains are designed for visualization, so we might as well put our eyes to good use! Some specific tools you might be familiar with are Venn diagrams (representing how sets intersect with each other) and the number line (representing the relative ordering of numbers, usually increasing from left to right).

As a subtler example, consider how in algebra we use variables to simplify our lives: it is clumsy to work with the equation $\text{number of oranges} + 6 = 3\cdot \text{number of oranges}$, so we often use a letter like $O$ to represent the number of oranges, and instead work with the more compact equation $O + 6 = 3\cdot O$ (which is easier to manipulate and solve). On the other hand, this "shorthand" notation takes some getting used to when first learning algebra: it's helpful to think of "unfamiliar abstract things" (like the letter $O$) in terms of familiar concrete things (like "the number of oranges").