Addendum: On Discovery and Inquiry


I appreciated some of the disagreement that got aired as a response to my last piece, on discovery. In particular, some told me that guided-inquiry or discovery really is more memorable than other forms of instruction.

Either because the stuff you learn from discovery is more meaningful (and hence more memorable):

Screenshot 2017-11-07 at 9.31.30 AM

…or because discovery involves surprise, and surprises are more memorable and lead to stronger learning:

Screenshot 2017-11-07 at 9.29.10 AM

I’m not satisfied with either of these arguments.

The thing about discovery activities is that the new idea — by definition of discovery — comes at the end of the activity. That means that kids are spending most of the activity thinking about stuff besides the new, often difficult, idea. It takes time to understand new ideas — to make them meaningful, to “own” them — and most of the time in a discovery activity is spent thinking about other stuff.

That’s certainly the case for the triangle angle activity that I critiqued in my post. While working on the activity, a student’s attention is drawn to many mathematical things — the angles, protractors, adding angles — and only very little of the time is spent thinking about what exactly a triangle’s angles sum to. (This is especially true if the idea is truly new to a student — they’ll only be thinking about the sum once they discover it, towards the end of the activity.)

It’s also true in the trapezoid/triangle area task that I shared. There was a ton of excitement precisely because my class hadn’t discovered the relationship between bases and area yet. That was where the joy was coming from — that also means that they were thinking about the discovered relationship for comparatively little of the time spent on the activity.

As I argued in the original post, that’s OK for me. It was fun and beautiful, and kids should have a chance to articulate slippery patterns and feel the pleasure of discovery. That’s part of math that I enjoy sharing with kids.

Anyway, that’s my response to the idea that discovery is more memorable because it’s more meaningful. Ideas are meaningful when you have time to get used to them, and that’s precisely what gets lost in a discovery activity.

As far as the idea that guided inquiry is surprising, and surprising stuff is more effective: why can’t you structure an explanation to elicit prior knowledge and surprise students? Aren’t explanations sometimes surprising? I think they can be.

Of course, how to craft effective explanations — that surprise and really engage students — is not easy, but it doesn’t get any easier if we don’t talk and write about it. That was part of my argument in Beyond “Beyond Explaining.”


This is all theory, though. What happened in class today, after the weekend, after the memorable discussion on Friday?

I ask them to find the area of a trapezoid and…it’s like Friday never happened.

Wait what?

Hold on what do you mean the same as a triangle?

Could we go over this again?

The only kid who remembered how to find the area of the trapezoid — and I promise this is true, and not just me making up details to annoy advocates of discovery — was the kid who had connected Friday’s lesson to a formula that she once knew.

No guys, it’s the sum of the bases times half the height.

This is sort of surprising and disappointing. Friday’s class was so good! And nearly everybody was involved in the inquiry/discovery/discussion. It felt wonderful and it was fun.

That class, for me, was discovery that’s about as good as it usually gets. And yet it failed to stick over the weekend.

And yet this isn’t that surprising. The kids didn’t get a chance to practice the idea on Friday because we spend the class time uncovering some super-cool math. Kids need practice to remember ideas, and discovery takes a long time. This is just how it goes.

But if it’s not surprising, it’s also not disappointing. It was a lot of fun, and everybody was involved. It’s not what my class is like every day, and it would probably frustrate kids if it were.

So, at least this time, anecdote matches argument. And since we started practicing finding the area of trapezoids today,  it’s getting a lot more meaningful for my kids.

12 thoughts on “Addendum: On Discovery and Inquiry

  1. What woud have happened if you would have given them all the dot examples after the weekend and asked: write down everything you remember about these patterns? As a retrieval practice. Would that be more close to (returning to) where they were after the dot activity?

    Liked by 1 person

  2. For me, inquiry is a part of the process, and it doesn’t work if I start and end with it. The essential part of guided inquiry lesson is the reflection where students have a chance to gather and communicate thoughts about what just happened. Then coming back to it the next day and sharing some reflections, practicing new skill. Then possibly my favorite no or find a mistake. Then some more application/practice. After some time, getting back to the skill/concept and applying it in light of the new skills.

    This year I got some students in my gr5 who I taught two years ago in gr3. We did geometry/compass constructions/islamic art mathart inquiry project. Kids were identifying and building polygons and discussing properties as we went. Two years later, many of them still rember our conversations and referred to their gr3 experiences when we started talking quadrilaterals.

    Inquiry is not a lesson though or a one-off activity. It is a part of learning process that involves exploration, problem solving, reflection, direct teaching and distributed practice.

    I often think about language learning: in order to really learn a new word or a language structure, you need to encounter and use it multiple times in multiple contexts. I don’t think one lesson of any kind (only discovery, only direct teaching, only practice) would be enough to make anything truly meaningful. You noticed that as your students started practicing, it is getting a lot more meaningful. I think that while inquiry creates often a fuzzy picture in students minds that still needs work and refining, it can build an intuition on which later I can hang practice and direct instruction and to which I can refer when I revisit the topic a few months later.

    Liked by 2 people

    1. Mark Chubb wrote a blog post called “Don’t skip the close” iirc. Debriefing your findings should be a substansial part of any inquiry lesson. If you don’t have time that class period you need to do it the next day or shorten the activity to fit it in that day, but don’t skip it!

      Liked by 1 person

      1. I love the idea of “don’t skip the close,” especially for an inquiry activity. But here’s where I’m still at:

        * It takes time thinking about a new idea to properly understand it.
        * Being meaningful and memorable is a function of properly understanding a new idea.
        * Because the new idea often comes at the end of an inquiry lesson, there is often less time spent thinking about that new idea compared to other activities.

        Now I totally see why a lesson close needs to be a substantial part of an inquiry lesson — because you need to spend a lot of time thinking about that new idea, and comparatively little time was spent thinking about the new thing during the inquiry.

        My argument in these posts hasn’t been “therefore inquiry is bad.” Just “therefore inquiry is going to be less effective than a hypothetical replacement that calls on kids to think about the new thing throughout the activity.” Personally, I find inquiry fun and important for sharing the nature of mathematical creativity. I just balance that with my knowledge that it tends to be less effective than alternatives.

        Practice and time is important for learning — we all agree — but there are other activity formats besides inquiry, and they involve more time-on-new-topic.


  3. I really appreciated the opportunity to engage in a dialogue about this so I hope you don’t mind me pushing back again.

    I don’t think students forgetting on Monday what they learnt on Friday shows that inquiry-based learning isn’t memorable and that it doesn’t lead to deeper learning. I think it does show that it takes time and practice for students to internalise new ideas. If they didn’t remember the formula for a trapezoid, were they able to use the thinking from Friday to figure it out? For example, if shown the dot page from Friday, I’m guessing that arriving at the same place would have been much faster the second time around. They need to do the same thinking but the connections happen faster the more they do it.

    I also think it’s important to distinguish between discovery learning (where the students discovered a pattern) and inquiry learning (where they go on to investigate the limits of the conditions under which that pattern holds true). It’s in undertaking this inquiry that students would get to use the generalisation that they had ‘discovered’ thereby practicing it repeatedly with the type of focus that you are aiming for. But rather than just practising applying a formula, they have a bigger purpose: to explore the limits of the generalisation. And that’s what I think makes inquiry learning more memorable.

    I’m also going to push back on a comment you made about wanting to be the teacher who can be trusted to answer questions. I’m the teacher who often (not always) answers a question with another question: “What makes you say that?” “How could we find out if this is always true?” Sometimes my students find this frustrating but I’m ok with that because I want them to get used to the feeling of not-quite-knowing while also building their capacity to be self-sufficient.

    Liked by 2 people

  4. What you said makes perfect sense to me. If you don’t practice a new idea to reinforce, you lose it. (Especially if there is a weekend involved!) I wonder about “Ideas are meaningful when you have time to get used to them”. I think ideas are remembered when you have time to get used to them. Meaningful ideas can be a lightning bolt or a slow discovery. But either way, without reinforcement, the idea is lost, no matter how meaningful it was.

    This post gives me a lot of food for thought. I’m using the Illustrative Math curriculum most days this year and I love it. I love the inquiry, I love the engagement, I love the hands-on activities. But I do think I need to build in more time for practice. If I had a 90 minute class each day, they would get enough following the curriculum as-is. But since we have 45 minutes, I need to tweak something to fit it in.

    Liked by 1 person

    1. Is Illustrative Mathematics an inquiry curriculum? I’m using it in 8th Grade and I see a lot of things that I like — example analysis, practice, conceptual questions, explicit teaching of subtle things, etc. I agree that I need to supplement more practice, but kids always need more practice.


  5. Just a note for myself: what I wish I had tried, during the heat of things on Friday, was pausing everybody and asking them to find some paper to do some personal writing to synthesize everything that we had discussed. I’ve seen this a million times from Marilyn Burns, Kristin Gray and David Wees, but I still haven’t figured out how to make this work well in my classrooms. This is perpetually on my list of things to figure out, and I haven’t figured it out yet. Need to find a chance to experiment with this.


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