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Correcting Aunt Sally
"Please Excuse My Dear Aunt Sally" is known to the majority of my students. They recite it with vigor. Unfortunately, they often do not know how to apply it. There are usually three major misunderstandings:
  1. Please (students are taught this means parentheses) means any grouping symbol. The appearance of nested grouping symbols stops many students from continuing to solve the problem.
  2. Students do not understand that operations are performed from left to right
  3. Addition/Subtraction and Multiplication/Division are of the same level. One does not do all the addition and then all the subtraction. Rather as we move from left to right we perform addition or subtraction as we encounter it.
Because so many are familiar with the mnemonic, it helps to clear this up immediately upon introducing the topic. There are two ways to introduce the topic. One is to give the rules and then apply those rules; the other is to build the rules.The first method is the most often used, so perhaps the second method may be more effective for students when the first didn’t work!
To build the rules, I suggest a whole class discussion broken up by group work performed by pairs of students.
  1. Begin with problems where one must only worry about addition and subtraction. (Practice Sheet provided) Have small groups simplify the first expression (put subtraction before addition to challenge PEMDAS misunderstandings head on). After groups have an answer, give them the correct answer. Ask those that got it wrong to check to see if they can find their mistake; if not they should check with a group that got the answer right to begin with. Once all are comfortable with this, have students complete that group of exercises (2 and 3) and check all answers.
  2. Sprinkle in some parentheses and repeat the process. Emphasize that parentheses need to be removed before proceeding. Also, they sometimes change the outcome and sometimes do not. (Compare #1 and #2 with #4 and #5.)
  3. Continue with some multiplication and division. Again, have the leading problem present division before the multiplication (#7) and repeat the process as in #1.
  4. Sprinkle in parentheses and fraction bars as we did in item #2 above.
  5. Discuss with students the fact that an expression such as 32 cannot be used in calculations since the exponent is notation indicating an operation. They should replace all such expressions with their values before using the number in calculations.
  6. Now try some simple examples to combine operations. Repeat the process we used in item #1 above. Problem 12 on the sheet is provided to give your students a minute to explore. Once again, let them try; then give the right answer and have them resolve any conflicts.
  7. You may need to remind students about the distributive property at this point since students may see it as an exception to the order of operations. Explain that it is used to remove the parentheses when simplification within the parentheses is not possible.
After students have completed the sheets and checked answers, have them write rules for the order of operations in their own words. Take these up and read them carefully. Return any with errors and have students correct; meet individually with students who have serious misunderstandings. As a follow up, have them compare the rules they have written with the ones in the textbook.