The saying “Please Excuse My Dear Aunt Sally” is not actually a letter of excusal, but an acronym for remembering the order of mathematical operations in mixed operational math problems and equations. In their article titled “Order of Operations: The Myth and the Math,” Jennifer M. Bay-Williams and Sherri Li Martinie discuss the assumptions and mathematical basis behind the order of operations. Mathematics can be confusing when it comes to determining certain liberties and rules within the system. For example, the commutative property allows us to add numbers in any order, but other operations require us to solve from left to right to get the correct answer.
Are we teaching children the rationale behind the existence and necessity of these ordered operations, or are we simply providing them with facts to follow, thus limiting their problem-solving skills and hindering creative approaches within mathematical principles? Bay-Williams and Martinie debunk the misconception that the order of operations was arbitrarily determined long ago. This fallacy suggests that the rules and operations in math were decided upon through mutual agreement, which is not how mathematics works. In reality, the sequence in which we solve mixed operation math problems is based on mathematical logic, particularly when considering the meaning of the numbers involved and the implications if the order of operations were altered. Another misconception addressed is that the order of operations is inflexible.
Students can still approach problems in different ways by using the commutative properties of addition or multiplication. It is also recognized that these orders can be taught conceptually. Multiple operations math problems can represent real world problems and can be explained using a conceptual basis. For example, a teacher gave her students a problem involving stacks of coins and numbers of those stacks. The students concluded that adding before multiplying incorrectly created more stacks of coins, which is illogical. It is important for students to grapple with operation problems and understand why the order is necessary, as it helps them see the sense in it. The article emphasizes the importance of teaching conceptually to ensure that students comprehend the task. To aid in the instruction of these math concepts and procedures, meaningful tasks should be encouraged.
By teaching students the why behind math concepts, they can become more proficient in multiple operation equations and problems. I personally found the content of this article insightful, as it emphasizes the importance of understanding the meaning behind mathematical concepts rather than simply memorizing facts. Often, math is viewed as a subject of rules and facts with only one correct answer, similar to the scientific field. When I learned about order of operations, it was presented as a strict rule to follow without questioning its reasoning or approaching it conceptually. However, by teaching students the reasoning behind how and why a problem works (or doesn’t work), we are fostering critical thinking skills rather than mere rote memorization.
The article discusses a word problem that is also an order of operations problem. It establishes credibility by citing a section of the common core standards which emphasizes the importance of finding entry points and planning a solution pathway for children. This aligns with our classroom learning of approaching math problems from multiple perspectives and understanding that there are various ways to solve them. Although some methods may be easier or faster, there are still multiple ways to reach the same conclusion. We have also explored different models for addition, such as the set point model, strip diagram, and number line.
This text illustrates the use of different visual aids and algorithms to solve math problems. These include the lattice algorithm and left-to-right algorithm for addition, which utilize partial sums. Additionally, the equal additions and Austrian algorithms are used for subtraction. These models and algorithms showcase the various approaches to solving addition and subtraction problems, and can be used interchangeably for the same operations. For instance, when adding 256 and 189, one can employ multiple visual models and algorithms to solve the problem. Similarly, when subtracting 189 from 256, different methods can be applied to reach the same result.
The article correctly points out that in the PEMDAS acronym, multiplication comes before division and addition comes before subtraction. However, it is important to understand that these operations should be performed in order from left to right as they appear in the equation. Some confusion arises when people mistakenly assume that multiplication should always come before division, which leads to performing the operations out of order. Interestingly, in other countries like Kenya, students are taught that division actually comes before multiplication. This is still true in problems involving multiple operations.
This does not mean that multiplication and division are commutative with each other, but it means that multiplication should not be done before division if the operation comes after in the problem. In conclusion, Bay-Williams and Martinie firmly believe that students should not be taught mathematical quick facts, but should instead be taught the concepts behind the steps and processes of math problems, especially in this case for finding solutions in problems involving multiple operations. This aligns with what we have been studying in Survey of Mathematics regarding understanding the operations and processes behind basic math problems, diving deeper beyond the surface of numbers and facts. If we can apply this learning approach, we can not only learn math more effectively but also learn all subjects more comprehensively.
