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Order of Operations: The Simple Solution

One of the topics in modern Arithmetic that students struggle with is the order of operations. By “order of operations” I mean that order in which operations are to be performed in a complicated Arithmetic problem. Here’s an example:

Boethius Teaches Arithmetic
Boethius (475-526 AD) teaching a student Arithmetic.

5 × 3 + 2 × 4 – 2 ÷ 2 =

Let us solve this problem by moving left to right, performing one operation at a time:

5 × 3 = 13 + 2 = 15 × 4 = 60 – 2 = 58 ÷ 2 = 29

If we were to enter this answer on a test, it would be marked wrong.

This is why many students struggle with the “order of operations” in Arithmetic and Algebra.

Now, as with all things, when the professional educators find a topic that students have a problem with, they see it as a chance to make money. So they invent all kinds of “tricks” and “tips” to solve this problem, recommend tutoring and special programs, on and on.

This problem, however, is caused by one simply lesson the students have failed to learn, and it can be easily fixed.

Understanding Terms

In modern schools and study programs, very little attention is given to memorizing rules and definitions in Arithmetic and Algebra. Most students, if asked, would not even be able to explain what Arithmetic and Algebra are. They don’t learn definitions.

This failure in modern math programs is the cause of this problem with “order of operations”. If students simply understood one concept, all of this trouble would go away.

That concept is the term in mathematics. Here is the definition:

A Term is an expression not united to any other by the sign + or − .

Not knowing that definition right there is the cause of all of the trouble with “order of operations”. A complex expression, where there are a number of different signs, is made up of simple expressions, called “terms”. Before we can solve any such problem, we must first divide it into its terms, using this definition as our guide.

So, when we look at a complicated Arithmetic or Algebra problem, we need to ask, “How many terms does this expression contain?”

Let’s look at our example:

5 × 3 + 2 × 4 – 2 ÷ 2 =

A term is an expression not united to any other by the sign of addition or substraction. Therefore, in the example problem, we need to mark out the addition and subtraction signs because they divide the simple terms in this complex expression.

We see that this expression is made up of three terms:

5 × 3
2 × 4
2 ÷ 2

Because these are separate “terms”, they are to be treated separately. We are not to simply solve each operation from left to right in our problem because this problem is made up of three different terms.

Solution: The signs of addition and subtraction mark the different terms in an expression. Perform the operations within terms first, and then those between terms.

We can use parentheses to mark the terms and this will make the problem very easy:

( 5 × 3 ) + ( 2 × 4 ) – ( 2 ÷ 2 ) =

Now, we can see what this expression really says and the “order of operations” is very simple.

( 15 ) + ( 8 ) – ( 1 ) = 22

It’s that simple.

Conclusion

The problem students have with the “order of operations” in Arithmetic and Algebra does not require any fancy tricks or tutoring. The problem is caused by the teachers’ neglect to teach definitions clearly to the students. Not knowing what a “term” is, students cannot understand how quantities are expressed in mathematics, and this leads them into a thousand problems, which modern educators see as an opportunity to make money.

By simply learning the definition of a term, and learning to divide a complex expression into its terms using parentheses, all of this trouble is prevented–and the student actually understands the math rather than copies solutions.

In the Classical Liberal Arts Academy’s mathematics courses, we focus on teaching students the language of mathematics so they can clearly understand these things and avoid these modern problems. If you were confused about the order of operations and found this explanation helpful, you’ll likely find our mathematics courses much better than modern math courses.

For more information on our self-paced, online math courses, visit the Quadrivium page in our course catalog.

God bless your studies,
William C. Michael, Headmaster
Classical Liberal Arts Academy

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