Evaluating Expressions - Misconceptions

Evaluating expressions is one of the foundational skills in algebra. It’s the bridge between arithmetic, where students mostly work with numbers, and algebra, where letters represent unknown or variable values. Because of this transition, many students encounter difficulties. Their mistakes often stem not from carelessness but from misconceptions that develop as they try to apply arithmetic rules to algebra without fully understanding the differences. This article highlights four major misconceptions students face when evaluating expressions: misuse of the order of operations (PEMDAS), misunderstanding the negative sign, errors in substitution, and combining unlike terms. For each misconception, we’ll look at why it happens, how it shows up in student work, and how teachers can guide students toward correct understanding.

Misconception 1: Misusing PEMDAS

Perhaps the most common struggle students face is remembering the correct order of operations. One way of teaching the order of operations is through the use of acronym PEMDAS (which stands for parentheses, exponents, multiplication, division, addition, and subtraction). Students often memorize the acronym but apply it incorrectly. A widespread error is believing multiplication must always come before division, and addition must always come before subtraction.

• Example of the misconception: 20 ÷ 5 × 2 = 20 ÷ (5 × 2) = 20 ÷ 10 = 2

• Correct approach: (20 ÷ 5) × 2 = 4 × 2 = 8 The misconception comes from taking PEMDAS too literally, as if the “M” must always precede the “D.” Students who memorize rules without context tend to follow the acronym in order, which overrides their reasoning.

• Teacher strategy: One of the most effective ways to address this is by explicitly teaching that multiplication and division are on the same level of priority, as are addition and subtraction. Instead of chanting “PEMDAS” in order, some teachers now use “PEMDAS with arrows,” showing that M and D go left to right, as do A and S. Demonstrating side-by-side problems where the wrong interpretation gives a different result helps students internalize why the left-to-right rule matters.

Misconception 2: Misunderstanding the Negative Sign

Negative numbers are another area where students’ reasoning often falters. Many learners think of the minus sign as something permanently “stuck” to the number, rather than recognizing it as part of an operation.

• Example of the misconception: If x = -5, then students asked to find -x often write -5. They see the variable as carrying the negative sign permanently.

• Correct approach: The actual calculation is -(-5) = +5. The negative sign in front of x is a separate instruction to negate whatever value x has. This misconception arises because students sometimes learn to read negative numbers as “a number with a minus attached.” That idea works fine in arithmetic, but in algebra, it can lead them to misapply the sign rules.

  
  

• Teacher strategy: Teachers can reinforce that the minus sign is an operation, not a permanent “sticker.” One simple classroom activity is substituting different values into -x and comparing: If x = 3, then -x = -3. If x = -3, then -x = +3. By showing students that the result changes depending on substitution, you emphasize the role of the minus as an operation, not part of the variable’s identity. Encouraging the use of parentheses, like writing -(x), also helps clarify what’s happening.

Misconception 3: Substitution Errors

Substitution is where students begin to apply arithmetic rules to algebraic symbols directly. Errors happen when students forget to follow proper operations after inserting a value for a variable.

• Example of the misconception: If x = 3, then in the expression 2x + 5, some students substitute incorrectly: 2 + 3 + 5 = 10. They treat “2x” as though it means 2 plus x, instead of 2 multiplied by x.

  
  

• Correct approach: The right evaluation is (2 × 3) + 5 = 6 + 5 = 11. This misconception is partly due to notation. When students first see algebra, the lack of a multiplication sign between the coefficient and variable can be confusing. In arithmetic, multiplication is always written explicitly, like 2 × 3. Without the “×,” they may assume addition instead.

  
  

• Teacher strategy: When first introducing substitution, it’s helpful to go back to the expanded forms. Writing 2 × x before shortening it to 2x ensures students understand the multiplication connection. Another strategy is using color coding or highlighting - show students that 2x means the product of the coefficient and variable, and substitution means replacing x with a value, not tacking it onto the expression.

Misconception 4: Combining Unlike Terms

The last major misconception appears when students evaluate expressions involving more than one variable. They sometimes think that substitution allows them to combine variables together in ways that break the rules of algebra.

• Example of the misconception: If x = 2 and y = 5, students may evaluate 3x + y as: 3(2 + 5) = 3 × 7 = 21. They incorrectly add the variable values together before multiplying, treating 3x + y as though it were 3(x + y).

  
  

• Correct approach: The correct evaluation is (3 × 2) + 5 = 6 + 5 = 11.

  This misconception often happens because students are still transitioning from arithmetic operations to algebraic expressions. To them, seeing letters next to each other looks like something that should be grouped, and they overlook the fact that multiplication and addition are distinct steps.

  
  

• Teacher strategy: One of the best ways to address this is to draw attention to structure. Teachers can use brackets deliberately to highlight the correct grouping: 3x + y really means (3 × x) + y. By practicing with clear grouping, students gradually stop misreading the expression. Having students verbalize their steps - “three times x, then add y” - also strengthens their understanding.

Pulling It All Together ...

These four misconceptions - misusing PEMDAS, misunderstanding the negative sign, making substitution errors, and combining unlike terms - are deeply connected. They stem from students overgeneralizing arithmetic habits into algebra without adjusting for new symbolic meanings.

For teachers, addressing these errors requires a mix of:

1. Clarity in instruction: showing explicitly how algebra differs from arithmetic.

   
   

2. Multiple representations: using parentheses, number lines, or verbal explanations to reinforce operations.

   
   

3. Deliberate practice: giving students side-by-side “wrong way” and “right way” problems so they can see why certain approaches fail.

   
   

4. Encouraging reasoning: asking students to explain their steps out loud or in writing, which reveals where their misconceptions lie.

When students grasp that evaluating expressions requires structure and attention to operations, they gain confidence in algebra. Teachers can use misconceptions not as failures, but as opportunities to deepen understanding. By explicitly addressing these common mistakes, educators equip students with the tools to approach algebraic evaluation with accuracy and clarity.