I think of myself as a non-discovery math sort of teacher, but every once in a while I find myself asking kids to discover stuff. I recently did this in my geometry classes, with a dot-paper area activity.

I really like dot paper. On dot paper you can make precise statements about area that typically generalize nicely to non-dot scenarios. The same basic relationships apply, it’s just easier to see them in a dot world.

My idea was to share this picture with students, and ask them to find the area of each shape. (This is the version that I marked up under the camera in class. None of the purple or blue ink was in the original.)

All of these shapes have the same height — 4 — but they otherwise differ. The first is a rectangle that has area 8. Next was a parallelogram: also area 8. Then a trapezoid. Many students came up with the idea of drawing a line to split it into a triangle and a rectangle — together they have area 8. This didn’t occur to every kid, though, so there was a good deal of neighbor-asking and chat to get the hang of those trapezoids.

Spoilers: every shape here has area 8! There’s something else they share too: the sum of their top and bottom bases is always 4. (The weirdest case is the triangle, that has a bottom of base of 4 and [arguably] a top base of length 0!)

So that was what I wanted my kids to come to notice and articulate. And I wanted it to be fun, and feel like they were discovering something new.

Not because I think that learning is more effective when kids discover something new, or that they’re working on their discovering skills or anything. Just because I think it’s fun for them to uncover patterns. It’s a cool part of math, and I’m trying hard to share more of the cool parts of math with my students, along with working on their skills and knowledge etc.

Here then is my take on discovery in math class:

- It can be fun to discover cool stuff about math.
- It takes longer for kids to understand something via discovering, and doesn’t really confer a learning advantage.
- But if the activity is relatively brief and I can spare the time, why not? I want my students to think math is fun and cool.

This is my unprincipled take. I like discovery for fun and color in class, and I don’t feel the need to aim for 100% efficiency in every second of my teaching.*

* *Whether you feel such a need — or need to feel such a need — probably varies a lot depending on your school, administration, students, etc. My guess, though, is that the vast majority of teachers don’t feel this need, and probably are correct in this feeling.*

Lots of discovery activities are uninteresting to me. Though I absolutely love the practice exercises in *Discovering Geometry*, the discovery activities largely leave me cold. Here’s an example of the sort of thing I’ve fallen out of love with:

My issue with this activity isn’t that it’s discovery. It’s that it’s not such a cool pattern (most kids have seen it before taking a geometry class), so discovering it isn’t as surprising or fun. The activity takes a while — do it once, check, do it twice — and all that is probably to protect against the risk of incorrect measurement, which is another tedious aspect of this discovery activity. And, at the end of all this, what cool math have you uncovered? Relatively little — just a sum. You don’t see any new relationships or geometric structure that guarantees that the triangle will have such a sum.

Why bother with all this? I’ve decided that this sort of discovery activity isn’t much use for me. But that’s not a principled objection against discovery — it’s just that I don’t think this type of activity is worth it.

Speaking of “types of activity,” I think it’s fair to categorize this angle sum activity as **discovering something easy to articulate**. Check an example, check an example, check an example, woah it’s always the same. You don’t uncover the geometric relationship in this activity. I think that’s part of what makes it not much fun and sort of tedious.

I think **discovering something hard to articulate**** **(I’m not quite sure what to name this) tends to be more fun, more cool.

To illustrate this, here’s the conversation that went along with my “area 8” activity in class.

I began the conversation with a prompt to my students: what do all these shapes share? how do these shapes differ?

I called on James first.

*James: They all have heights and bases of 4.*

I said that I didn’t entirely get that, and asked if anyone also saw that. Robin came up to the board to point. She also subtly refined Jame’s claim.

*Robin: ***A lot** of them have a common theme that either the height is 4 or the base of 4.

Then Liam chimed in to make it precise and accurate:

*Liam: The height is always 4. The bases are different though.*

*Luiz: Yeah, the bases are either 2 or 4…or no they vary. Sometimes it’s 1, 2, sometimes it’s none.*

Then Sara chimes in. She started articulating a generalization — she was WORKING HARD to try to articulate some sort of generalization. Her first one wasn’t entirely accurate though. I loved how she put it as a question.

*Sara: Wait, does that mean that any shape that has a height of 4 has an area of 8?*

The class and I (and Sara) agreed: this just could not be true. (Luiz says: well you could have a base of 1000.) Jess tried to get clearer about what was special about all these shapes.

*Jess: No this is just because they are all parallelograms and…wait the third one is sort of confusing. What is that?*

[Insert a minute of discussion about which of these are trapezoids and which are parallelograms.]

After this, I decided that we wouldn’t be able to restart the conversation unless I summarized and took some notes. So I wrote some notes on the board.

This time, though, I decided to take a heavier hand to draw attention to something really cool. The kids hadn’t noticed it yet, and I wanted to make sure that they did.

I wrote the numbers that you saw in purple ink in my image. I wrote the top and bottom bases.

*Sara: So couldn’t you say that it’s base 1 plus base 2, times the height and that’s the area. Like for shapes in general.*

Very close! There are some gasps and agreements.

*Samantha: So wait does that work for everything?*

*Sara: Yeah that’s what I’m asking.*

*Me: That’s a good question. I’m trying to find some dot paper.*

*Joe: Wait so does it?*

That question just hangs there for a second. And here’s a choice I could make. I could act coy, refuse to answer, and insist that the thinking here come from the students.

But then you get this weird dynamic in class where kids never know if they’re getting a straight answer from the teacher or not. I don’t like that dynamic. I like it when kids ask questions about math, and I like that they can get a straight answer out of me. And would they spend more time thinking about this cool relationship if I answered that question, or if I refused to?

“The answer is yes, sort of.”

GASPS.

*Sara: Does it have to have parallel lines? Does it have to be a trapezoid…wait does a parallelogram count as a trapezoid.*

Good thinking, Sara!

While there’s thinking kind of just toppling out of Sara’s head, I’m searching for a blank piece of dot paper, because there’s something that I realize might help. I don’t want them to get too used to the area 8 case — that might lead to a false generalization, since Sara keeps on saying that it’s *double *the sum of the bases. (That’s true when the height is 4, but not when the height is something else.)

So I draw this:

We clocked in at about 10 minutes there. No question that this was not as effective as a worked example or something else more carefully designed for learning. But I wasn’t aiming for efficacy. I was aiming for those half-articulations, those gasps, that enthusiasm. And as long as I don’t come to worship those gasps and chase them exclusively, class will be a bit closer to being fun, cool.

**Addendum: **this follow-up post.

I wonder what effect the techniques have on students retention of the ideas. I’ve used a lot of discovery this year in the Illustrative Math curriculum based on my convictions that they will remember the ideas for longer (weeks or months instead of minutes or hours.) True? I guess we’ll see next year…

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I share many of your views on discovery, although, I must say, the discussion you record here would qualify as a home run lesson in one of my classrooms.

Anyway, I especially like the discussion of “fun” at the heart of your reflection, though. Messing around with patterns is as fun to a mathematician as messing around with offbeat rhythms or scales can be to a musician. There doesn’t have to be a “song” at the other end of the exercise. There doesn’t even have to be a purpose. Still, it’s these moments that most increase mental flexibility and make it easier to take creative risks.

Long story short: At the time of year when things are starting to feel like a forced march for many of us (“We have to meet topic X by date Y or the battle is lost”) you gave your students a chance to see how a lifelong mathematician might entertain himself (or herself). I commend you for it.

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I think the area lesson is worthwhile. Not for its ‘efficiency’, as you say, in quickly knowing how to find areas of shapes, but in allowing the students to behave like mathematicians, to look at the evidence, to form a generalisation, to progressively revise this with discussion.

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In some ways this seems like an activity with elements similar to a goal-free problem. Not going straight for the answer, but exploring the information in the patterns. Which may in a way be quite efficient.

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Or maybe not a goal-free problem, given the text ‘Find the areas!’ at the top.

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