Lesson Developing Your Visual Ability

This is an excerpt from the book Inner Algebra: How To Do Algebra In Your Head. You can read the full text of the book at hilomath.com.

As someone who is learning algebra, you are exercising new parts of your brain. You are learning to think abstractly and logically. That's valuable! As you get better at thinking in those ways, math becomes easier and more fun.

An important key is to be able to visualize. That basically means using your imagination in certain ways - purposefully seeing things in your mind's eye. Most good mathematicians have great visualization ability. Here's a secret: Almost anyone can learn it! This lesson teaches you how. Done correctly, it can help make all the math you do easier.

Getting Started

Think of a character you like from a comic, computer game, or movie. Close your eyes and imagine this character -- see them in your mind's eye. Even if it is a fuzzy or incomplete image, that's fine. Just get a definite visual impression of this character.

If you can do that, great! You already have some natural visualization ability.

If you could not conjure up this image just now, or you are not sure, don't worry. First, it is not necessary to be able to visualize in any great detail. Let's say the character you picked is a talking duck you liked to watch on TV when you were younger. If you can see just part of the face and bill for a split second, and then it's gone, you can visualize plenty well enough to start. So just focus on getting that.

Practicing will get you there. Here are some tips:

• Try something. If it works, great. If not, try another approach. What helps one person visualize really well is often different from what will help another person.


• It will probably help to choose a character you have positive feelings about. Think of a little child watching a cartoon, and suddenly a character the child really likes comes on the screen. What does that child feel inside?


• Visualize with your eyes open. Visualize with your eyes closed. Which is more effective for you?


• If it's not easy to visualize the character you picked, pick a different character, or some object that has meaning to you. For example, if you skateboard, imagine a skateboard, or last year's X Games champ flying on one.


• Find a real picture of the character, or draw one. Look at the picture. Close your eyes, and try to imagine that picture. Repeat. Use the picture as an aid.


• Bring different senses into it. Some people are "visual" people; they will usually be able to imagine things in their mind's eye immediately. If you are more of an auditory (sound) person, imagine this character talking or making a noise. If you are more of a kinesthetic, "feeling" person, imagine the character doing something physically active, and what they would feel while doing it.


• Move the location of the image. When people imagine something, they usually put it in a specific place in the space around or inside them. Some people envision it in front of them, or slightly up or to one side. It can also be located inside your body, such as inside your head. Try several locations to find what works best for you.


• Be willing to give it time. Work on it a little, and if it is not easily coming to you, do something else. Try again in a few hours or another day.


• Practice until you can consistently get a mostly complete image, and can hold it for a few seconds. When you can do that, continue on.

Visualizing Equations

Now we are going to try something with this equation:

Since this equation is so simple, you probably know the answer just by looking at it. But it doesn't matter if you don't. How easy or hard it is to solve does not matter -- we are going to do something else entirely with it.

Imagine that you have written this equation in big letters and numbers on a 3 by 5 card, like this:


Imagine that this card is floating in front of you, where you can see it. It's at a comfortable distance. You can move it around by imagining it. You want be able to see each individual symbol, 'x', '+', '2', '=' and '3', and their position in space relative to each other. Practice this until you can consistently see the equation on the card. It does not matter if you do it with your eyes open or shut. It can help to write the equation on a real 3x5 card, and glance at it as an aid.

By the way, you certainly do NOT need to discern it in great detail. If, in your imagination, you can recognize each of the five symbols, and get the order right, that's all that is necessary. In other words, as long as YOU can tell that the first symbol is an 'x', even if it does not really look anything like it, that's fine -- you are the only one who will ever see it anyway!

Now, take the image and move it around. Move it a few inches left, right, up and down. (Your head is motionless. You are getting practice at controlling where you put the image that you see mentally.) Again, practice this until you can consistently do it.

Now make the image of the equation bigger and smaller. You are zooming in and out. Make it twice its original size. Make it half its original size. Practice until you can do it consistently.

Remember, in all of this, you're not doing anything with the equation, mathematically speaking. You are not taking any steps to solve it. You are merely creating a mental image of it, just like you could imagine a car or a banana or anything. You are then moving that image around while keeping the image more or less intact.

Here is a summary of the steps above:

• Visualize the equation x+2=3 as if it was written on a 3-by-5 card and held in front of you. In your mind's eye, see it in enough detail that you can discern each of the five symbols and their order in the equation.


• Move the image around -- left, right, up, down, towards you (in), away (out) -- while keeping it whole and clear.


• Zoom in and out. Make the image of the equation half its original size, and twice its original size.

If you can do the above, and feel comfortable with it, do the exercises below. If you are having difficulty with it, you have some options. One is to continue forward -- it may click into place, as you do the exercises. Another option is to stop reading for a day. Tomorrow, practice the above three steps to refresh yourself, and to let you get more comfortable with it. When learning something new like this, giving yourself time to absorb it can be helpful.

Exercises

Pick three equations. You can choose them from a math book, or just make them up. Pick equations that you consider to be slightly challenging -- that is, equations that are complex and 'big' enough to be interesting, but not much bigger. (You are not going to have to solve them, so you can relax about that!) It's best to pick equations that are fairly different from each other.

When you've chosen them, do these exercises with each equation:

• Visualize the equation as if it was written on a 3-by-5 card and held in front of you. In your mind's eye, see it in enough detail that you can discern each of the symbols and their positions in the equation.


• Move the image around -- left, right, up, down, towards you (in), away (out) -- while keeping it whole and clear.