## October 23, 2024

As kids progress through math and concepts become more complex, they’re often confused by misconceptions created when they were younger. Today, we’re addressing common math misunderstandings that arise when students work with operations.

Let’s break down four widespread **misconceptions** around math **operations**.

Misconception 1: Adding always increases the value.

Misconception 2: Multiplying always increases the value.

Misconception 3: Subtracting always decreases the value.

Misconception 4: Dividing always decreases the value.

Let’s tackle each math misconception and talk about:

- Why these misconceptions are harmful
- How we can avoid creating them
- How to support students who already have these misconceptions

But first, let’s create some common language.

Misconception is defined by Merriam-Webster as a “wrong or inaccurate idea or conception”. When working with students, I like to frame misconceptions as mistakes in our thinking around an idea. It’s when we believe something is true based on a misunderstanding we have. As we learn more, we can revise our thinking and correct these mistakes.

**Misconceptions around the operations**

Misconception 1: Adding always increases the value.

Misconception 2: Multiplying always increases the value.

Misconception 3: Subtracting always decreases the value.

Misconception 4: Dividing always decreases the value.

**Why are these math misconceptions harmful?**

When students begin to work with integers and rational numbers, it can feel like all the rules they learned no longer apply. Their solutions don’t make sense and it causes frustration. These frustrations can turn into negative beliefs and mindsets around math.

This is not only challenging for students – but for teachers and caregivers too.

In reality, the understandings around these operations are based on misunderstandings that occurred and were internalized years before.

For example, let’s consider multiplication of fractions.

The solution to 1 x 2 is 2. The solution to ¾ x 2 is 1 ½.

But wait… 2 is equal to 2 and 1 ½ is less than 2. How can this be? If we’re multiplying, shouldn’t the product be the greatest number in the equation?

Similarly, adding negative numbers, like -50 + -50 = -100, challenges the belief that adding always increases the value. These examples show why the misconception of ‘always bigger or smaller’ doesn’t hold up with more complex numbers.

If I have a negative balance in my bank account, let’s say -$50, and then I overdraw my account by another $50. I now have -$50 + -$50 which is -$100. I added and my solution had a lesser value!

How confusing is that if we believe adding and multiplying create larger values, and subtracting and dividing create smaller values?

**How can we avoid creating this belief?**

When kids first begin to formally learn math, we focus on the natural, or counting, numbers (1, 2, 3…) and whole numbers – the natural numbers and zero.

Every time they add or multiply, the result is larger. Over time, they start to believe this is always the case. But this rule doesn’t apply to integers and fractions, which is why we need to make sure we don’t overgeneralize.

When I was in the classroom, I remember times when I explicitly said things like, “When we add, we end up with a bigger number.” This is true, but *only* when working with whole numbers. That’s the part I was leaving out.

The same is true for subtraction but vice versa.

So how can you help kids avoid this misconception?

- Tell them that the kinds of numbers you’re working with are called whole numbers. Say things like, “When we add whole numbers, we do end up with a bigger number. When you’re older, you’ll learn about other kinds of numbers and that won’t always be true.”
- Expose kids to integers and rational numbers. Yes, even in kindergarten! It can be as simple as having a number path posted that shows negative values. This way, kids visually see that numbers don’t start at zero.
- Avoid using key words. They’re problematic when it comes to comprehending word problems. They also reinforce this misunderstanding by associating words like “more” with addition.
- When teaching subtraction, avoid language around taking away. “While the idea of taking away holds true for whole numbers, it causes confusion when students begin to add or subtract integers. They don’t understand why we end up with a greater value when subtracting a negative number. Using language such as “separate” instead of “take away” helps avoid this confusion later.” (Eleczko, Faulkner, Russo 171)

**What to do when students already have these misconceptions**

When students already hold these misconceptions, explaining how they came about is key. Show them when their ‘rules’ hold true and when they don’t.

Let’s go back to our example of owing $50 and then owing $50 more. While referencing a visual number line, here’s what that might sound like.

- I noticed many of you were confused when we were adding negative numbers. I have a hunch it might be because we think adding means we end up with a greater value. Let’s take a look at what’s happening.
- When you first learned to add, you worked with whole numbers. If you had $50 and got another $50, you combined those amounts and ended with more money, a total of $100. (Show a leap of +50 on the number line and then another leap of $50.)
- Now let’s say you didn’t have any money but there was a pair of shoes you’ve been wanting and they’re on sale for $50. You’re starting with $0, you borrow $50 from a friend. Now, you owe $50. (Show a jump of -50 on the number line.) What if the same thing happened again? And now you owe another $50? (Show another jump of -50 on the number line.) You’re still combining amounts but because you borrowed money your solution is now less than what you started with.

These are abstract ideas. To help students correct their earlier misunderstandings, it’s helpful for them to see connections between the conceptual and procedural.

Another thing to consider is how we can avoid creating more misconceptions. Think about the ‘rules’ we teach students. Will they always be true? What happens if they forget a rule? Have we tied enough meaning to the procedure that they can make sense of what’s happening without needing to remember a rule?

**Conclusion**

Misconceptions around math operations are common. By using clear, consistent language when we introduce concepts, we can prevent these misunderstandings before they take root. For those students who already hold these misconceptions, providing context and showing them when their ‘rules’ apply (and when they don’t) can rebuild their confidence in math. Thinking about math holistically helps students gain an understanding of how mathematical ideas, and specifically operations, truly work.

There are many common misconceptions just like these, some of which we’ll explore in later blogs. If you’d like to learn more about them, I’d love for you to join my Structures of Equality (SoE) communities.

Eleczko, Faulkner, Russo. *The Fire & Wire Way*. 3rd ed., Lulu.com Publishing, 2024.