Section 19.4 Closing Ideas
One of the challenges for students is that fractions seem so different from the rest of their arithmetic. All of the additional "rules" that are put into place to do the calculation make it feel so much different than addition with integers or even decimals. Multiplication is in some ways the easiest because it's still just multiplication, and the top and the bottom are often separated by a bar to help visually separate the two pieces.
But there's a different challenge that comes with this. When the calculations are easy to execute, it's easy to think that you understand something when you're consistently able to get the right answer. Understanding is not the same as executing. Going back to the analogy that was introduced earlier, knowing how you to drive a car doesn't mean that you know how a car works. But with mathematics, many students are taught a mentality that the execution and the understanding are the same thing. Hopefully, as you've been working through this text, you're starting to adopt a different mindset.
It is often stated that strong math skills are important for finding good jobs that pay well. But do you think those jobs are about doing calculations? We have computers that can do that billions of times faster than any human can. So they clearly aren't just looking for people who can execute calculations. They want people who can think mathematically. In fact, there are many people who are only average in their computational abilities in those types of jobs. (Once again, they have computers to do the calculations for them!) But they are able to get into those positions because they know how to think mathematically. They understand how to reason through ideas and make connections.
As we continue forward in our journey, we're going to be looking at ideas that are going to get increasingly familiar. There's even an entire branch that focuses on basic arithmetic. Remember that the goal isn't to teach you how to do calculations. The goal is to help you to develop your skills in mathematical thinking.