In conversations with anyone from members of the math department to random strangers in coffee shops (all of whom have opinions about teaching) to family members, people often bemoan to me the idea that, “students these days don’t know things.” The complainant will often follow this statement with a list of content that students don’t know – how to simplify radicals, where India is, how to identify a complete sentence, etc.
And you know what, it’s sometimes true. Some students don’t know how to simplify radicals* (unless they learned in middle school). But I am only worried by what students “don’t know” when I am trying to teach them to solve a specific type of problem in a historically-identified efficient way. When I give students a rich, low-threshold, high-ceiling task (which by design means there are multiple ways to enter the problem and that students can take their thinking far), I get to see what students know. They can enter the problem and move through the more challenging questions at their own pace/using their own method. They can compare their method with others, argue about and defend an answer, and find another way to answer the same question.
Basically, the problem isn’t about students not knowing things, it’s about me not asking the right questions often enough. And accessing prior knowledge doesn’t (have to) mean trying to remember the CMP book that teaches the topic you’re about to address or writing a contrived word problem about food to relate something to students’ lives. Accessing prior knowledge can mean offering up a really good question** and letting students pursue it any way that makes sense to them (and they will!). When I allow students to do that, I get to experience all the things students know. Then, I can work on connecting all those ways and building up students’ problem-solving toolboxes.
p.s. I painted a slightly idealized picture of low-threshold, high-ceiling tasks. But it’s mostly accurate.
* And who the hell cares?
**Eventually, hopefully, I’ll figure out how to tap into students’ questions.