## Common Errors Versus Math Misconceptions

Misconceptions in mathematics are inevitable and often predictable, but teachers can use them to support deep learning by helping students recognize their misconceptions and revise their thinking. When educators encourage members of their classrooms to think of their initial mathematical solutions as drafts, they’re giving students the opportunity to “edit” their thinking—just as they would in the evolving drafts of an English paper. University of Delaware Mathematics Education Professor Amanda Jansen calls this approach “rough-draft thinking.”

Using rough-draft thinking allows all students to learn from one another’s misconceptions. Math misconceptions aren’t the same as math errors. What distinguishes a misconception from an error or misapplication is the root of the mistake; it is a student’s understanding of the concept or relationship that is incomplete or incorrect. In other words, it is a conceptual misunderstanding, not an error in calculation.

### What Is a Math Misconception?

A math misconception is, essentially, an incomplete understanding that we’ve developed from our math experience. Math misconceptions are also known as “alternative conceptions” and/or “intuitive theories.”

Here’s a first grade–level example that illustrates the difference between an error and a misconception:

**Common Error**: 28 + 47 = 76

**Explanation**: “I added 8 plus 7 and got 16, so I wrote the 6 in the ones place, and then added the 1 ten to 4 plus 2 to get 7 in the tens place.”

**Error**: Thinking 8 + 7 is 16, not 15

**Common Misconception**: 28 + 47 = 615

**Explanation**: “I added 8 plus 7 to get 15 and then added 2 + 4 to get 6. I wrote the 6 next to the 15 because it was in the next place value.”

**Misconception**: Not understanding place value

## How Applying Prior Knowledge Incorrectly Can Lead to Misconceptions

There are many reasons students develop misconceptions. In a previous post, I described how adults can say things in ways that create misconceptions. But another common path to misconceptions is when students attempt to apply prior knowledge to new concepts; these initial conceptions or preconceptions may be incorrect, but offer productive starting points for students to explore and revise their rough-draft thinking.

Think about a student who knows that when multiplying 15 x 4, the commutative and distributive properties allow them to create an equivalent expression such as 15 (2 + 2), which then allows for the operation to be split into two easy-to-handle partial products, (15 x 2) and (15 x 2).

When students begin to explore and make sense of division, it’s entirely natural and reasonable for them to think that 80 ÷ 4 might be expressed as 80 ÷ (2 + 2) and that this leads to the expression (80 ÷ 2) + (80 ÷ 2).

Although incorrect, this stems from a reasonable attempt to extend one’s understanding of multiplication to the operation of division. What’s important is that students are encouraged to try out such ideas and are given the space to discuss and debate whether such claims are true in every case.

In trying to explain and justify a claim such as the one above, students come to realize—often with help from their peers and certainly their teacher—that the distributive property does not always hold true when working with division.

As part of this process of explaining, justifying, and revising one’s thinking, it’s helpful to provide students with multiple representations through which to examine and explore the concept or relationship. In the division example, students might be encouraged to use base-ten blocks, draw array models, and/or set the problem in a familiar context to support their sense making.

The teacher might then use student-generated examples to lead a class discussion about why the distributive property holds true when partitioning the divisor (e.g., 80 ÷ 4 = (40 + 40) ÷ 4) but does not hold true when partitioning the dividend (because, unlike multiplication, division is not commutative).

This post has been adapted from *Recognizing Misconceptions as Opportunities for Learning Mathematics with Understanding*. To learn more about using student misconceptions to build deep conceptual math understanding, click below to download the whitepaper.

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