What Is Mathematics?

Teaching Math as a System of Patterns, Context, and Choice

Math is my jam and I am here to spread it. 

I saw this on a t-shirt once and I haven’t been able to stop thinking about it since.

Link

I have been thinking about the nature of math a lot lately - whether I am listening to one of my favorite math podcasts (Math Therapy by Vanessa Vikharia) or catching up on the latest issue of NCTM’s Learning & Teaching, or taking my online PreCalculus class at a local community college, it has been on my mind. 

Here is what I have come to realize about math:

1. Mathematics is a set of systems with its own language based on patterns and relationships

As a teacher of math, it is my job to provide experiences and opportunities for students to learn about the different systems, see the patterns and relationships, and help them make connections. My favorite way to do this is by using the Experience First, Formalize Later approach. Those familiar with Math Medic or Amplify (Desmos Classroom) may be familiar with this strategy. One of my absolute favorites is the Amplify activity on Composition of Functions. Students are first experiencing what composing functions means in a context they are familiar with - shopping. They have to make predictions, perform some calculations on which discount order gives the customer the cheaper price. Once they experience what composing two functions means, then they are exposed to the formal notation for composing functions. 

2. When we understand the system and its patterns and relationships we get to MAKE DECISIONS about how we approach and solve problems

As a teacher of math, I am tasked to help students build conceptual understanding AND knowledge of strategies that can be employed in different contexts. For example, when we understand fractions - WHY we need to have common denominators to add and subtract, WHY we multiply the numerators and denominators when we multiply, and WHY we flip and multiply when we divide - we can apply those ideas to more complex contexts. If you want to rationalize a denominator, you are really multiplying a fraction by “a well chosen 1”. Understanding that when you multiply something by 1, it does not change the value of the thing. If we are creative in how the 1 looks….. We can make magic happen.

3. Context matters - sometimes patterns and relationships cannot be transferred across different systems 

I cannot overstate this enough - context matters. And not the crazy word problem context where we arbitrarily make someone buy 12 watermelons, I mean the context of what numbers mean. For example, when we add 10 + 3 we get 13 in our base ten number system, but if we need to know the time 3 hours after 10:00 am, it is now 1:00 pm. Think about a linear relationship - when you double the input, y doubles - if you double the amount of hours you  work, you will double the amount you earn on your paycheck. However, for a quadratic relationship, when you double the input, you quadruple the output - if you double the sides of a square, the area quadruples. The context in which we are performing these operations matters. In other words, 1 + 1 is not always 2.

So What?

Understanding what math IS helps us figure out what is important when helping others experience and learn about it. My job is to help students learn about different systems in mathematics. My job is to help students build their fluency and strategic knowledge so they are empowered to CHOOSE how to approach and solve problems. My job is to help students build confidence in their ability to math. 

After all, math is my jam and I am here to spread it.