# Generalized Arithmetic

## Equal Sign

To help reveal your students’ thinking about the equal sign, ask them to solve this problem: “8 + 4 = ? + 5”. Most elementary school children solve problems like this incorrectly. Most often, they add 8 + 4 and put 12 in the box, or they add 8 + 4 + 5 and put 17 in the box.

Where does it come from? Analysis of students’ textbooks has shown that almost all instances of the equal sign are at the end of the problem. Children incorrectly infer that the equal sign must mean “get the answer” or “the total,” rather than “the same as.” Children are making a logical inference from what they have seen. However, this misunderstanding is not inevitable! When teachers systematically engage their students in solving a variety of problems with the equal sign in different positions and discuss the meaning of the equal sign, students are much less likely to misunderstand the equal sign.

Finally, understanding the equal sign matters. Even in middle school, many students define the equal sign as meaning “get the answer” and these students are much less likely to solve equations correctly. Doing the same thing to both sides of an equation doesn’t make sense if you don’t understand the meaning of the equal sign!

Go to the Apply page for ideas on how to help your students understand the equal sign.

## Properties of Operations

To help reveal your students’ understanding of the commutative property, ask them to add two numbers and then ask how quickly they can add the numbers in the opposite order (e.g., 5 + 6 and then 6 + 5). For older students, write an addition problem and sum on the board and ask students how quickly they can figure out the sum for adding the addends in the opposite order (e.g. 34 + 12 = 46; 12 + 34 = ?)

Some students do not use the commutative principle (the order you add in does not matter). Others use it, but cannot articulate it. Similar activities can be used to reveal their thinking about addition and subtraction as inverses (e.g., 34 + 12 = 46. 46 – 12 = ?).

Go to the Apply page  for ideas on how to help your students understand and apply properties of operations.