*Bits, ink, particles, and words.*

If you’ve ever learned logic, you know that jumping from one piece of information to another (through implication) isn’t always clear. You might remember something about if A implies B, then not B implies not A, and the like. What I want to do today is try and explore these concepts in a more visual way, which hopefully will let you remember these concepts with greater ease.

When students first start working in mathematics, units aren’t really something to worry about. Indeed, students begin with doing arithmetic, where there’s little need for extra fuss about units. Students get used to writing equations without ever thinking about units.

If there’s a bane to everyone’s existence in mathematics, the word problem seems to be it. These are routinely the most tricky problems to solve, and so many students find themselves understanding the material, only to stumble on this kind of problem. If you ask a student, chances are that they will say these problems are difficult because they aren’t as straightforward as a question that asks to solve for $x$ or to find the area of a figure. I don’t want to spend today debating whether or not it’s a good idea to have word problems, but instead I want to address the fundamental difficulty that I’ve found many students to have in this area.

One of the most common misconceptions I see while working with students in secondary school is the notion of an inverse. The idea isn’t too complicated, but the reason that I see students making mistakes with it is because they are in the process of learning about functions and it becomes a cognitive burden to think about these abstract processes such as inverses and other transformations. However, I firmly believe that giving students the right idea of how these different concepts fit together will help them navigate their classes with ease.