Lots of really thoughtful people got me thinking in their responses to yesterday’s post. Check the comments and Twitter for a sample. Three are really sticking with me.

Anna Blinstein:

I think that my only addition or caveat to your post, Dylan, is to push back a bit on the goals of math class. If the primary or only goal is remembering/applying mathematical content knowledge, then your post makes complete and total sense – we should probably use discovery sparingly; it is helpful as a motivator (basically, the intellectual need and wonder categories you listed) and maybe helps some students remember some ideas some of the time. But, if one’s goal is to teach students to think like mathematicians, then I don’t know of a better way than having them engage in the process of doing math consistently and frequently while also seeing models of what this might look like and getting feedback on their efforts and ideas. I don’t think that anyone would argue that a discovery approach is the most efficient method of transferring knowledge, but for me at least, that’s not the primary goal.

Dan Anderson:

(my answer)

Avery Pickford:

These all hit me pretty hard, and I’m questioning a bunch of what I wrote yesterday. I’m going to try and reframe my argument and see where this goes. Call me out if I’m totally off track here. There are two issues I see. Does discovery lead to better learning of content, and does discovery learning teach students to think like mathematicians and acquire future knowledge for themselves? The second question is the one I want to tackle.

I think the answer is yes. First, an example.

**Introducing the Unit Circle**

I recently introduced the unit circle. Students started by constructing right isosceles and equilateral triangles to derive the values of sine and cosine at a few angles. I chose a discovery approach for this lesson because it had students practicing relevant triangle geometry, seemed manageable, and the much more didactic alternative I was imagining sounded boring. The activity felt successful; every group found the value they were looking for, we followed up with a brief discussion, and then groups derived the rest of the unit circle.

I was surprised when, three days later, many students failed to produce a similar proof on an assessment. I don’t mean to argue that discovery never works, just that in this instance it came up short, and that doesn’t feel like an isolated example to me.

Contrast: after that assessment, I gave students this image:

I told them that this was a new unit circle, with angles at 15, 75, 105 and etc degrees. I gave them the values of sine and cosine at 15 degrees and asked them to figure out as much additional information as they could.

I would argue that this activity also asks kids to think like mathematicians. It’s much more subtle, in that there is no grand reveal of the unit circle at the end. But it’s still valuable thinking. And, more importantly for me, I find that more students have a chance to access a task like this, because they all have similar background knowledge at this point. This task also involves acquiring knowledge, and can be metacognitive in thinking about strategies that are useful for acquiring knowledge. And if I have to pick between a discovery activity introducing the unit circle and a demanding task asking students to reason with what they’ve learned, I prefer this one. Not by a ton, but I do.

Discovery often meets ambitious goals of mathematics learning. I think that, for many students, my unit circle exploration did that. But it didn’t land for everyone. And, based on my experience with that activity and similar activities, it’s too often the same few kids who are left behind.

I don’t believe that discovery has a monopoly on teaching students to think like mathematicians, and I don’t believe that every lessons has to involve discovery in order to teach kids how to acquire knowledge themselves. I’m at a 3.5 on Dan’s scale, which I take to mean somewhere around a third of my lessons involve some element of discovery. I would like to nudge that number upward a bit — but only if it means I’m doing discovery well, for every student. I would rather focus a small amount of energy on these goals but do that well than try to do it every day and possibly undermine those same, important messages.

I hear the argument that it’s important to teach students to think like mathematicians and acquire knowledge for themselves. But I want to avoid being dogmatic about using discovery as a means to do so. Instead, I want to focus on doing discovery well when I use it. Maybe I’m just coming up short in my knowledge and skill with that particular pedagogy; I’d love to hear folks sound off in the comments with the practices that have proven useful for them with these types of lessons. But in the meantime, I’m going to try and meet all of the above goals with as many different means as I can.