Have you ever wondered if we are giving them a slight misconception when we teach equations? As shown in the picture, these equations result in a number. Most linear equations we teach also equal a specific number such as 2x + 7 = 15. We have them solve it and they know the value of x is 4 but the idea the value can change becomes a bit more difficult for linear equations such as 2x + 7 = y because it does not equal a constant.
Most people see the end result of the equal sign when they go shopping. Once they've checked out, the register totals all the prices and they pay.
How do we get students to go from looking for a set numerical answer to thinking about equality being a two way street. I often teach it only one way such as 2x + 7 = 15. I seldom even write equations as 15 = 2x + 7 or even as 2x+7 = 6x - 11. This becomes a problem when my students take trig and have to work both ways when they are proving trigonometric equations.
According to one article I read, over 70 percent of middle school students use a running total when solving order of operations rather than understanding the whole process. An example might be 4 + 2 + 8 = ___ + 5. Many of these students will treat the blank as a place to write the sum for the left hand portion of the equation. They solve it this way 4 + 2 + 8 = 14 + 5, not understanding the left side equals 14 while the right side should equal 14 so the missing number should be 9.
The equal sign is a relatively new invention dating from the 1500's. Before then, mathematicians used words such as yields, or gives. Now the equal sign has several different meanings:
1. It gives the result of a calculation such as 4 + 6 = 10
2. Decomposition such as 18 = 6 + 12
3. Equivalence between two expressions such as 3 + 9 = 2 x 6
4. The equal sign can indicate equivalence such as in 2/3 = 4/6
So it boils down to three different meanings for the = sign. The result of a calculation, decomposition and equivalence. Do we teach all three meanings or do we assume they understand the variations according to the context?
Let me know what you think. I can tell you I need to work harder on conveying all three variations to help my students become more mathematically fluent.