Section 10.4 Closing Ideas
There's an internet math meme that makes its way around social media every now and then. There are a few versions of it, but they all look something like this:
People will argue for hours over whether the correct answer is or Some people will insist on the "rule" that division and multiplication and done left-to-right, but others will note that the implied multiplication is usually given priority. For example, in writing \(5 \div 2x\text{,}\) there's virtually no context in which we would separate the 2 from the \(x\text{.}\)
But all of that arguing misses the point. The "answer" is that the person writing the problem did not communicate effectively. The person who wrote the expression did not use notation in a clear manner. And that's an important lesson. Mathematics is not about arcane rules, but about communicating ideas.
One of the ongoing challenges for math courses is that students can keep advancing forward with a weak foundation. They can often do well enough to pass the new material while there are weaknesses from previous sections that remain unaddressed. And this can continue for a while, sometimes multiple years, until there's a certain moment where things suddenly make no sense at all.
Sometimes, students will get to that spot and try to brute force their way through it. It is not often successful from the educational perspective, though with enough effort students they may still be able to "pass" the material (even if just barely).
The first part of this book has been focused on shoring up a number of key algebraic ideas and concepts. This last section is an important step towards true algebraic fluency. As soon as you are able to "see through" all of the notation and break down complicated expressions to their more basic components, you have access to a much wider range of algebraic thought processes. Going back to the language analogy, this is the point where you cross over from speaking in broken sentence fragments into the early stages of fluency.
The next major step in forward development is contextualized practice. It's not enough to simply see these algebraic ideas in isolation. They need to be brought into perspective through the lens of other mathematical ideas. And that is where the college level math course will take over.
The rest of this book is about backwards development. In other words, it's about finding and filling some of the earlier holes that create problems for students. In a real sense, we will be working backwards from concepts found in high school algebra down to elementary school arithmetic. The content is full of a number of simple ideas that are sometimes missed by students as they're coming up through the educational system. And those are sometimes the concepts that help to support higher levels of mathematical thinking.