Mathcurmudgeon has a great post on whether right answers are the most important thing in mathematics education. Go read it. Highlight was this visual

My thoughts:

I am absolutely guilty of telling kids that the right answer is not most important, and I agree that it can be problematic. Students are unclear when right answers are important, and when they aren’t (doing questions as a class? homework? tests? modeling tasks? warm-ups? all different.) I’ve also seen students disengage when I say the right answer isn’t important. That’s what they were looking for; if that’s not the point they don’t know what is. It also contradicts the reality that, although there are exceptions, in much of math class and much of the real world, the right answer really does matter, just the process and the way you find it matter too. I like this line from his post:

What we should be saying is “The right answer is vitally important … so important that we also want students to explain the method and how we all know the answer is correct; they must be able to detect an error if it occurs and describe how to fix it so that the solution IS correct.”

I think there’s an awesome point here that will make more sense to students, while still structuring math class in a way that incentivizes students to pause and make sense of math, and give life to this idea of arguing about mathematics and critiquing others’ reasoning

That said, I want to be careful with Mathcurmudgeon’s logic. I think there’s enormous value in problems that don’t have a right answer — and make that explicit from the outset. Nat Banting has a great post on this topic, writing questions that have no one “right” answer, but promote metacognition and discussion about mathematical concepts. Below is a great example:

Fill in the blank with the number that makes this equation as simple as possible. Explain your choice. Once you’ve explained your choice, go ahead and solve the equation. Show all work.

I think this is great. The question has a low floor, a variety of answers that promote mathematical reasoning, and promotes discussion. Most important to me, it gives students feedback right away. If the number they chose makes the equation hard, they then have to solve a hard equation–and if they are confronted with a new number they hadn’t thought of, they can try it right away and compare it to their own.

This is the heart of what I’m trying to get at with my series of posts on problem design–the fundamental idea that connects all of this is to get students to think about and make sense of mathematics, instead of looking at it as a series of answer-getting steps to be memorized and regurgitated. And, in my view, teaching students those habits of mind is the sum of great problems that create opportunities for exactly that thinking.