Our geometry unit was drawing to a close and I felt like I had given them a good range of experiences with area and volume. We counted squares and we counted cubes. We derived formulas, but spent a lot of time digging into how the formulas represented what we had learned about the shapes. All along I kept dodging any attempts to get me to tell them "how to do it" by posing questions of my own: what are we counting? How is that like the way we estimated the cubes in the coffee cup? How would you build that shape from cubes? I was like a ninja ... you couldn't pin me down. And more and more students were responding with "Oh yeah, it's like layers in a cake / it's like levels / it's the area on the top times the height." I resisted giving them a formula for volume, letting it just be part of our conversations, until late in the process when I showed them this slide as part of a summary discussion.

I felt good. I felt certain that this was just summarizing an idea that they had internalized, that I had heard them using the idea when they worked together, and that they got it. So imagine my surprise when, as we were reviewing for our test on area and volume, students in each of my sections said at some time during the process, "You never gave us the formula for volume."

... There are times, I fear, when the best response I can come up with for the current teachable moment is "Are you kidding me?" Had I completely misread their level of understanding? Did they not get it? You don't just memorize formulas - you figure things out! What had we been doing for the past two weeks, for the whole year? Are you kidding me?

Now it is true that we did have a formula sheet for area. I guess I thought that there are different formulas for different shapes, and I wanted to have a page that summarized all of them. We had reasoned them all out for ourselves, and I didn't really expect them to re-derive the formula for the area of a circle on their own. But the volume of a prism or cylinder seemed different. How could they understand the concept if they needed the formula?

I must have been channeling Fawn Nguyen, who treats student input with such respect and reverence, because it suddenly occurred to me - this is a great opportunity to talk about this. So I wrote on the board -

And I asked "How many of you have ever come to me this year and asked me to tell you how to do something, but instead I said 'I think we better take another look at this'?" (About 90% of the hands went up). "And how many of you wanted to punch me?" (About 90% of those hands went up - and fortunately most of them laughed too). "So talk to the person next to you - what do you think about formulas? What are the pros and cons of having or not having formulas?"

Here's what they said:

more complicated there are questions, sometimes confusing

non-chaos chaos - scrambling

like cooking with a recipe like cooking on Chopped - either way, you just want the cake!

anthill with a system anthill on fire

hike with a plan just walking

The students are reading

So what do you think? Was this a failed unit? Formula or No Formula? What I really got out of this day was that sincerely asking the students what they think told me a lot more than I expected it would, and turned a stressful moment into a learning one.

The next day, during the test, the student who started the whole thing by asking for the formula seemed to be struggling. He asked me "When you find the perimeter of a triangle, do you have to divide by two?" Ah. With that and the volume problems, I encouraged him to think back over some of the things we had done in class, and he remembered finding one level for volume and the perimeter looked like it was making sense ... ah ha! A victory for thinking it through. And then afterward I saw that his surface area work was a jumble - "Yeah, my tutor and I did base times height ÷ 2 *2 ... or something like that." I asked "So were the formulas a help or a hindrance?" He said "Oh the formulas totally helped." Ah. Well...one step forward...

I felt good. I felt certain that this was just summarizing an idea that they had internalized, that I had heard them using the idea when they worked together, and that they got it. So imagine my surprise when, as we were reviewing for our test on area and volume, students in each of my sections said at some time during the process, "You never gave us the formula for volume."

... There are times, I fear, when the best response I can come up with for the current teachable moment is "Are you kidding me?" Had I completely misread their level of understanding? Did they not get it? You don't just memorize formulas - you figure things out! What had we been doing for the past two weeks, for the whole year? Are you kidding me?

Now it is true that we did have a formula sheet for area. I guess I thought that there are different formulas for different shapes, and I wanted to have a page that summarized all of them. We had reasoned them all out for ourselves, and I didn't really expect them to re-derive the formula for the area of a circle on their own. But the volume of a prism or cylinder seemed different. How could they understand the concept if they needed the formula?

I must have been channeling Fawn Nguyen, who treats student input with such respect and reverence, because it suddenly occurred to me - this is a great opportunity to talk about this. So I wrote on the board -

**Formula No Formula**And I asked "How many of you have ever come to me this year and asked me to tell you how to do something, but instead I said 'I think we better take another look at this'?" (About 90% of the hands went up). "And how many of you wanted to punch me?" (About 90% of those hands went up - and fortunately most of them laughed too). "So talk to the person next to you - what do you think about formulas? What are the pros and cons of having or not having formulas?"

Here's what they said:

**Formula***No Formula*more complicated there are questions, sometimes confusing

non-chaos chaos - scrambling

like cooking with a recipe like cooking on Chopped - either way, you just want the cake!

anthill with a system anthill on fire

hike with a plan just walking

*Formulas are useful and important, but they are hard to memorize*

I could figure out area without a formula

Formulas help you remember how

You have to know how in order to make the formula

If you learn the idea, you don't need a formula

You can't be given the formula right away

With a formula, you can remember it better, use it outside

Formulas are the easy way

With no formula, you learn to understand faster

You learn more yourself with no formula, and it stays longerI could figure out area without a formula

Formulas help you remember how

You have to know how in order to make the formula

If you learn the idea, you don't need a formula

You can't be given the formula right away

With a formula, you can remember it better, use it outside

Formulas are the easy way

With no formula, you learn to understand faster

You learn more yourself with no formula, and it stays longer

The students are reading

*Call of the Wild*, and one student compared formulas to the dog's cushy life at the start of the book, and no formulas to the kind of thinking and adapting he had to do in the wild. I really liked that.So what do you think? Was this a failed unit? Formula or No Formula? What I really got out of this day was that sincerely asking the students what they think told me a lot more than I expected it would, and turned a stressful moment into a learning one.

The next day, during the test, the student who started the whole thing by asking for the formula seemed to be struggling. He asked me "When you find the perimeter of a triangle, do you have to divide by two?" Ah. With that and the volume problems, I encouraged him to think back over some of the things we had done in class, and he remembered finding one level for volume and the perimeter looked like it was making sense ... ah ha! A victory for thinking it through. And then afterward I saw that his surface area work was a jumble - "Yeah, my tutor and I did base times height ÷ 2 *2 ... or something like that." I asked "So were the formulas a help or a hindrance?" He said "Oh the formulas totally helped." Ah. Well...one step forward...