We here at Math Mistakes are always happy to share thoughtful writing about mistakes and student work.

First up is **Nicora Placa**, whose Bridging the Gap is one of my favorite new blogs. She asks, “**Is a Careless Error Really Careless?**” Here’s the headline news from a 1982 experiment:

However, as he continued his analysis of the interviews, he realized there was another explanation.

Students who incorrectly answered the questions were doing something that made sense to them.Their intuition was to place the multiplier next to the letter associated with the larger group. Although incorrect, it was meaningful to students.

I’m a bit ambivalent about this. I worry that coherent explanations of mistakes are often post-facto explanation of intuitions in disguise. But I have very little evidence for my claim that doesn’t come in the form of extensive surveys about exponents.

One of my favorite things about her blog is how well (and simply) annotated it is. She’s added so many things to my reading list. Go check her out.

Next up is **Evelyn Lamb** with a nice piece about a university-level mistake. There is a precise mathematical definition for the terms “closed” and “open” as they describe sets of numbers, but blah blah anything I could do to explain this would be worse than the way Evelyn does, so just go and check out her piece:

I had underestimated the power of the English language to suggest mathematically incorrect statements to my students. In mathematics, “open” and “closed” are not antonyms. Sets can be open, closed, both, or neither. (A set that is both open and closed is sometimes called “clopen.”) [ … ] But in English, the two words are basically opposites (although for doors and lids, we have the option of “ajar” in addition to open and closed). My students used their intuition about the way the words “open” and “closed” relate to each other in English and applied that intuition to the mathematical use of the terms.

Maybe we can change that vocabulary? Is there a better way to describe those sets, at least during students earliest exposure to these concepts?

Finally we’ve got **Michael Fenton** who posts about an interesting interaction he had with a student who completely nailed a set of rational exponents questions and was baffled by some square roots:

He writes lots of interesting things that are worth thinking about, including a sketch of how he thinks learning rational exponents might go better. Go check him out.