Why is this mistake so enticing, and how might you help students avoid it?

[Compare: http://mathmistakes.org/?p=396]

Thanks again to Anna Blinstein for the submission. Follow her! Virtually! Not literally!

  • We don’t teach why multiplication distributes across addition, and we talk about a power to a power means you multiply, so it seems very hard to distinguish between this mistake and when you can distribute. Nice to have a habit of reading comprehension when teaching symbolic stuff. “So what does this mean?”

  • This is one of the most common mistakes I’ve seen. I’ve asked students about this, and one thing I’ve heard them say is that they learned that parentheses means do it to each thing. Whether or not a previous teacher actually said that, if the only problems that students see when they first learn about parentheses is problems where you are distributing multiplication over addition, then it is reasonable for them to reach this generalization on their own. So I think it would be helpful that when students first learn the distributive property of multiplication over addition, they also see problems involving parentheses where there is no distribution. Require them to think about it from the beginning. Secondly, I think we should refrain from referring to “the distributive property” and talk instead about “the distributive property of multiplication over addition” and “the distributive property of exponentiation over multiplication”. Students can explore why those two properties hold (and how they’re related) but why there is no distributive property of exponentiation over addition. Yes, it takes longer to always include those operations, but I do feel it helps to avoid this common mistake.

    • Yirmin

      Also with Excel it is very easy to show example and jam in real numbers to show how results will differ when you do it the wrong way… Because sadly with very limited numbers you can do it the wrong way and it still gives the right answer.