When should variables be introduced? I debated this for many days when I learned that my daughter was highly gifted. I asked myself if I should concentrate on basic arithmetic, reasoning, etc. I went back and forth between first teaching her to add two and three digit numbers and introducing variables simultaneously with those concepts. The key here is that my daughter could already add single digit numbers and some two digit numbers.
I tried various strategies to teach the concept and use of variables. First, I explained the idea of an equation. I made it simple. I simply said that an equation is when you have two things that are equal. I taught her how we write an equation using standard notation. I used simple things like 5 = 5, 1 + 2 = 3, etc. I then showed her examples of things that are not equations. I used inequalities like 4 <> 3, 4 <> 1 + 2, etc. The key was to present a ton of examples. The next step was a fun one. I taught my daughter how to solve simple linear equations using M&Ms. Yes, you read that correctly. I used candy. We typically give our daughter some sort of dessert after dinner. So, I figured it may be fun to use dessert to teach elementary math. I got a bag of M&Ms and a small bowl. I set 1 M&M on the table and 2 under the bowl -- making sure she did not know how many were under the bowl. I then put 3 M&Ms the other side of the table. I asked her to tell me how many M&Ms would have to be under the bowl so both sides of the table had the same number. It should not shock you to learn how quickly she figure it out since I told her she could only eat the M&Ms under the bowl if she got the right answer. M&M algebra became a daily favorite of my daughter after dinner. I just made sure she never had too many! We played this game for about a week. Once she was getting the answers quickly, I took out some paper. I set up an M&M problem and then wrote down the corresponding equation on a piece of paper. I put an empty box to represent the bowl. I set up a number of examples asking her to tell me which of the M&M piles corresponded to what number and what corresponded to the empty box. I then asked her to tell me the number needed in the box to make the equation true. The next step was easy. We replaced the empty box with letters like n, x, etc. I used different letters and moved back and forth between empty boxes and letters always asking her to set up an equivalent M&M algebra problem. A few days later, variables became second nature to her. She understood they were just meant to represent the number we don't know. Clearly, I let her eat the M&Ms when she got the right answers.
I know. Some parents are going to argue that I shouldn't have used candy. However, you can be careful with the amount of sugar and still make the learning process fun. After all, we all know kids love candy. This may not work for your kid for medical or other reasons, but there is always something you can use in place of candy. Just make sure your kid loves it.
Let me get back to the original question of this post. When should variables be introduced? I now believe they should be taught as early as possible. The concept is only marginally more abstract than that of a number. Why we wait until pre-algebra is beyond me. In fact, Stanford's Education Program for Gifted Youth first teaches the idea in K and 1st grade. It uses many different examples in many different situations. The upshot of this early introduction is that 2nd and 3rd grade EPGY students can solve an impressive array of equations and understand how to translate word problems into equations and vice versa. For instance, one of the standard questions in EPGY's 2nd grade final exam is to solve a system of equations like the following:
m + n = 5
m - n = 1
This is asking to find two numbers 1 apart that add to 5, and this is precisely how they are taught to solve the system. They do not look at it the same way that algebra students do. They learn the role of variables and what the problem means.
I hope this post gives you some creative ideas to teach about variables and equations. I will write about how to teach other concepts in future posts.