Change is everywhere. Beyond the sudden change of children's growth and maturity, there is the slow change of aging. Everything seems to be in a constant state of flux. As Heraclitus famously said, "you can never step into the same river twice." because the water is always changing.
This concept of change has pondered and baffled philosophers of old. These philosophers spent countless hours pondering change. Here's the thing: Mathematicians are always observing truths and using patterns to predict the future. This is increasingly difficult when one considers that everything, even the patterns we observe, are changing. What do we do when the patterns are in a constant state of change? Yet we don’t always see them change and when faced with the task of pausing the change to calculate it, we fail. At least the ancients did, because they didn’t have the right tools yet. This is why Parmenides was famous for exactly the opposite of Heraclitus : nothing changes. He stated that since you can’t quantify the change at any given instant, it wasn’t change. Which is a short walk to “all is god”, a heretical belief that has since been disproved. So how do we wrestle with change?
This is why learning the relationship of dependent and independent variables with respect to each other is so important. This is why we spend weeks discussing the slope, rate of change, or a line on a graph. This concept of change is everywhere. To start to understand change in the moment, we observe change over a large period. In pre algebra, we calculate the average speed from my house to the grocery store, in algebra we observe the effects of different rates of change on linear equations, and in calculus we finally learn how to calculate change in the instant. This mental shift is essential to jump from the observed to the theoretical and mystical world of mathematics. This mystical world is where God has revealed the deeper truths, which we continually search out, like the King in proverbs.
If you’re starting to wane on your excitement of mathematics this year. Remember this, it is working towards something great and if you don’t like the problem you’re working on, don’t worry, it’ll change.