Starting and Ending with Applications

What can be done to interest a student in a topic in math? I think you show them how the things they do can be related to it. What can be done to make a student interested to continue to do math? You show them how to do something they've always wanted to do by using it. For example, let's take trigonometry and very simple vector applications (which a student would see in any physics class anyway). Trigonometry is a subject that people tend to hate, so I'll choose that one.

A lot of students these days play video games and computer games. One of the classic archetypes of such pastimes is the game from the first-person perspective. In a virtual world, such as the ones created by these games, in order to move right, you press the 'right arrow' button or key. In order to move forward, you press the 'up arrow'. However, some people have gained an edge over other players and seem to be able to move faster due to a trick. See, if they press both at the same time, they can travel diagonally significantly faster than they'd be able to otherwise. Woe to the person trying to flee from such a canny adversary in this game! This works because of vector addition and the rules of right triangles. If you want a game to remove this possible abuse, then how can you change it? The introduction is to use simple equal vectors (meaning the player travels at a 45-degree angle with respect to the direction their avatar faces). Students exposed to trigonometry need a solid background in right triangle geometry, and they can apply that here. It scales up into trig by saying that it doesn't make a whole lot of sense to be able to sidestep as quickly as you can run ahead, and in order to keep the total speed constant, you need to have some function that behaves periodically to monitor it.

The closing topic for trigonometry is to allow the students to try to do something they've wanted to do since you stepped into their lives--pelt you with water balloons. The ballistics of physics can be reduced to a few simple equations and students get a lot of practice solving for specific variables in computing ballistic trajectories. You can approach the problem by having students describe the qualitative behaviors they'd expect from throwing a ball at certain angles to hit a target. Then, you can show them the basic tools they need (they should not be expected to derive ballistic equations unless they've had significant exposure to calculus) in order to use what they've learned to try to hit you with some fun projectile. Some serious set-up is required to do this, since a means of reliably launching a projectile with specific force is sometimes difficult to come by. The trigonometry is in this in that they have to establish a specific angle required for the strength of the launcher, and achieving a simple expression for this requires the use of trig identities.

Physics teachers may complain that you're using up some of their tricks, but motivating a student to do something is a difficult task at times. It's possible, though, if you're willing to get your feet wet...

-Mark