Monthly Archives: August 2014

Rational Expressions – Watts Up


Flipped Classroom – Back in My Day

It’s Sunday morning, and my younger son is awake and wants to watch his favorite cartoon. I’ve already recorded all of the episodes and can ask him which one he wants. I can then start playing it, and even skip through the first thirty seconds of commercials so that it starts at the beginning of that catchy theme song.

This method of watching TV is not unfamiliar to many of us, but we can all remember a time when we could only watch a TV show when it was on, during a designated time slot, on a specific channel, and hopefully with a functional “rabbit-ears” antenna. (Some of you can go back to an even earlier time, but I don’t want to extend this part of the analogy too far before I make my point.)

Before we were teachers, we were all students, and I’m sure we can all instantly create a “back when I was in school” story. Oh, the menial tasks we were required to do. And my mind naturally drifts to math class. We had to calculate square roots, copy notes from an overhead, solve systems of equations, find inverses of three-by-three matrices – all by hand. (Our calculators only had four functions and one memory slot.)

We can certainly tell our students these stories, but only to the point where they understand what we are saying, like a narrative from a historical figure. But we can’t expect these same students to fully appreciate the experience of carrying out these tasks. And why should we? We have calculators, websites, and tablet apps that take care of all of this for us.

So should we stop teaching these skills altogether? No. Should we only direct students to the appropriate sites and apps to carry out these tasks? No. The key is balance. While it’s important to be able to solve certain problems quickly, using a calculator or other device, it is also important to understand what that device is doing, and what the end result actually means in terms of the original problem.

Let’s start with simplifying radicals, only because that concept was brought up to me this week. I propose that the concept should be taught by hand at first, so that students understand why the original radical and its simplified form are actually equal. After some practice, we can show how to perform the same task with a calculator. The question many teachers ask is, when would they ever need to do this by hand after they know how to use a calculator? The answer – variables. You can always create an “unsimplified” radical with variables, in which case the calculator is practically useless, and the old skills must be used.

Variables appear in the real world more often than we think. Sometimes we don’t have all of the information, and we need to find a temporary solution that we hold onto until the required information becomes available. Sometimes we want to create a rule, or formula, that we can use over and over, with different sets of given information. They can appear in almost every type of problem, including the radicals, systems, and matrices I mentioned earlier.

So let’s balance the old and the new. Do things by hand for appreciation, and in preparation for situations where the calculator can’t help. But don’t abandon the calculator because it “does the work for us.”

Times and technology are changing, and we need to embrace those changes. And on the other side, we also need to help students understand how technology works so they can decide when it is or isn’t helpful. There are times when we can choose technology to get us to the answer faster, but sometimes we just have to go “old school.”

To get back to my original story: While I’ve been writing this, I’ve had to help my son by skipping through commercials, and getting him to different episodes of the same show, all by clicking a few buttons. But I can’t help but think of the day when I will take him to the movie theater for the first time, simply to appreciate the experience. None of the technology we have will allow us to simplify that situation. The movie will start at a certain time in a certain room. It will only play once, and we will not be able to pause or rewind or start over. The lines will be long, the popcorn will be chewy, and the soda will be flat. And we’ll have to sit through all of the previews before the movie actually begins – just like it was back in my day.


Rational Expressions – Dinner Spinner


Rational Expressions – Exterior Angling


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