Wednesday, June 14, 2017

\$167.36 On the Nose

From estimation180, Day 161:

What's the value of all the coins in the bowl?

Before you read on, take a moment and come up with an estimate.

Andrew was gracious enough to provide us with the receipt, so Rich and I decided to use the prompt as Act 1 of a 3-act task.  It would provide the kids with multi-digit addition, subtraction, and multiplication practice, and since the bank takes one-tenth of the total amount as a fee for non-members, we would also receive some formative assessment information on how the students thought about decimals and place value.

We set the kids up in random groups on whiteboards, and asked them first for estimates. Nicole, the ILA teacher next door, poked her head in.  She asked what was going on, and one of the kids explained that they were estimating how much money was in the bowl. After a few minutes of thought, she started looking around for a scrap of paper.  Finding none, she pulled a tissue from a nearby tissue box, wrote something down, and handed it to me.  I folded it up and put it in my pocket, distracted by all the activity in the room as the kids finalized their estimates and began figuring out how much 1/10 of 519 quarters, 898 dimes, 719 nickels, and 917 pennies was worth:

Emptying my pockets at the end of the day, I came across Nicole's estimate:

What began to fascinate me, what I wanted to know, wasn't how she came up with the number, but why, having been asked for an estimate, she came up with something so exact!  Not \$160, or \$170, or \$200, but \$167.36.  We've been playing around with estimation for years now, and we're continually encouraging the kids to choose friendly, round numbers as estimates, numbers that tell about how many or how much, not necessarily exactly how many or how much.  But we've met with reliable, obstinate resistance.  I looked back at some of the pictures I had taken of student work, looking for a record of their estimates, and while I did see estimates like \$300, \$40, \$50, and \$60, I also saw \$63.12, \$5.57, and \$312.10.  Pointy numbers.  Precise numbers.  Numbers that spoke of exact amounts.  Not round numbers.  Not in the general vicinity numbers.  Not numbers that spoke of about how much.  Why?  Is there something hard-wired into our human nature that, when presented with a task like this, makes us want to be exactly right?  Not close enough, but closer than any of our classmates?  Have we been so conditioned by the "Guess How Many Jelly Beans in the Jar"  challenges that we treat every estimation task as a chance to win a prize?
"Why so exact?  Were you trying to guess the exact amount?"
"No,"  she explained,  "I was trying to estimate.  But I guess in my mind they're the same thing."
A few days later, eating lunch in the teacher's lounge...

 A leftover party favor from a week-end birthday party.
...four of my non-math teaching colleagues found themselves unwitting participants in an experiment.
"I want everyone to estimate how many gumballs are in the container."
The group was willing to cooperate, and within several seconds one piped up:
"Are we going to count them?  We have to find out who won."
I quickly got up and searched for a piece of paper and a pencil to write down the quote.  She had, on her own, without any suggestion from me, injected an element of competition and challenge into the task.
After a few minutes I asked for their estimates.  I received 3 pointy numbers, 84, 74, and 78, and one round number, 190.  (This teacher had estimated 192 but rounded down to 190.)  Although I wasn't so much interested in their reasoning, they all wanted to explain their thinking, and carefully listened to one another as they each shared their strategy in turn.  I explained my purpose in asking.  I was curious, I explained, why the three hadn't chosen round numbers as estimates.
"The answer is never a round number!" one explained.
That statement gives me a clue as to what may be at work here.  During these kinds of  estimation tasks, I'm asking kids to engage in sense-making, not answer-getting.  Maybe the line between the two is blurred, but it's there.  \$167.36 is an answer, not an estimate.
Answer-getting is stubborn.  I know this is true, because my teacher's lounge colleagues were just dying to know, couldn't wait to find out exactly how many gumballs were in the container.  They couldn't let it go:

gumball1 from Joe Schwartz on Vimeo.

Can you?

1. Fascinating!

One thing this makes me think about is some of the statistics books I've been reading about lately. In statistics, we don't just predict a single number, but we also try to quantify our confidence in the guess. And we also often report our predictions (is that what an estimate like this is?) with a range of responses.

Sometimes, before a 3act, I've asked my students to make their best estimate, and they give a range. Some kids think that's cheating, but it's really a thoughtful response to uncertainty.

What if we asked students to report a range of responses and their confidence in them? Or even if we asked them to provide a range of ranges?

"I am 100% confident the bowl is worth between ... and ..."
"I am 75% confident the bowl is worth between ... and ..."
"I am 60% confident the bowl is worth between ... and ..."

1. I like this idea a lot. This reminds me that I've seen Graham do a 3-act, and instead of asking for individual estimates he gives a range, and then asks how many students have estimates within that range. But I always forget to do that. I've had students give me ranges, and I encourage them to settle on a single number, but now you've got me thinking that maybe I shouldn't. Not a too low and a too high, but more squeezed range.

2. I imitate Dan a lot. After the estimate, ask for a too low and a too high.

Now I'm wondering how good at estimating students get with daily practice. Paging Mr Stadel...

3. We ask for too lows and too highs as well. I see that as being what Michael is calling the "I'm 100% confident that the bowl is worth between...". I like that way of explaining a too low and too high. But let's say the gumball estimate was 82. You might say instead, "Between 80 and 90." That's different to me then a too low and too high.
What I'm thinking now is that I've been working with estimation180 for a long time, but I still have lots to learn!