*(This all happens in one math class. Most of the students are away on a field trip and there is a very relaxed atmosphere. One student stands in the back of the classroom bouncing a badminton shuttlecock on a racket.)*

**Student #1:** “This is [Name]. He’s on exchange from Denmark and he’s going to all of my classes this week. This is his first day at [School].”

**Several Students:** “Hi!”

**Teacher:** “Today I’m going to be introducing vectors.”

*(She explains what vectors are and then starts showing us a proof of something but she makes mistakes so it takes a long time and we get confused. In the meantime, the student in the back loses control of the shuttlecock and it very narrowly misses the exchange student’s head. She apologises, retrieves the shuttlecock, and keeps going.)*

**Teacher:** **still writing out the proof** “So then we simplify that and get C equals…”

*(She pauses, looking confused.)*

**Student #1:** “C equals C. We did all that and just proved C equals C.”

**Teacher:** “No. That’s shouldn’t be right.”

**Other Students:** “We proved C equals C! We didn’t prove anything!”

*(We laugh uncontrollably.)*

**Teacher:** “Well, anyway…”

*(She abandons the proof and moves onto other topics.)*

**Teacher:** “These vectors are parallel. These vectors are orthogonal. That’s just another word for perpendicular. Let’s look at an example. Can someone tell me what the relationship between these two vectors is?”

**Student #2:** “There’re orthodontal vectors.”

**Teacher:** “That’s right, but the word is orthogonal. Now just so there’s no confusion for our exchange student orthogonal means parallel but orthodontal has to do with getting braces.”

*(She draws a pair of orthogonal vectors and a set of teeth on the whiteboard and labels them appropriately. This is the first and only thing she explains clearly in the entire lesson. We are all laughing. The exchange student sat silently for the entire 90-minute lesson and probably had a terrible first impression of American education.)*