## Check For Understanding

There's some dynamite material in the comments of the previous post, particularly from Alex, Andrew, Frank Noschese, and Steve Peters.

For anyone interested in curriculum design, please consider: which of these is the better question to ask a class?

1. How much time will it take to fill up the tank?
2. How much water will the tank hold?

### 16 Responses to “Check For Understanding”

1. on 23 Feb 2010 at 3:08 pmScott Elias

I’m going with how much time to fill the tank. More variables to play with in that question. And you still have to find out how much the tank holds without asking it outright.

Pencils down.

2. on 23 Feb 2010 at 3:08 pmJenny

I’m an elementary teacher and I don’t usually jump in on these because the math is over my head. It’s been years since I took math at this level (high school, college) and while I read a lot about math and teaching at my level it doesn’t help me with the sorts of questions Dan asks.

That said, I think the first question is better. It seems like asking the first question may (only may, I think) push the second question in while the second question could stand alone. Is that completely off base when you go to solve this?

And my #2 pencil tip broke so take this for what it’s worth.

3. on 23 Feb 2010 at 3:13 pmAndrew

Such a teacher to turn a question I just asked around to the whole class.

I think the time question reduces to a simple proportion problem. Time how long it takes to fill 1/10th of the tank, multiply by ten. (Not to understate the importance of proportions, but there’s more there.)

Asking the volume question forces the students to seek out additional information. They need a flow rate of the water, or some way to see the dimensions of the container (in which case the water is irrelevant).

It seems the, “What more information do I need?” moment is an integral part of the WCWDWT method. From what I can see, the time question just doesn’t get there.

It’s hard to argue with Frank’s lesson, though. If you’re seeking an understanding of sampling, error, and prediction, the time question is superior. Removing the level of abstraction volume introduces allows students to see the sources of their error more clearly. Adding the layer makes them think about what’s missing. It depends on what you want them to learn.

4. on 23 Feb 2010 at 4:18 pmAlex

Every kid at some point (hopefully) has grabbed a hose that was turned on and covered most of the nozzle with their thumb to create a smaller yet much stronger flow of water. As a kid I can remember wondering if this meant more water was now coming out of the hose. It’s this natural curiosity that makes “how much time will it take to fill up the tank” a far more valuable leading question. Everyone can participate, and most will want to.

5. on 23 Feb 2010 at 7:22 pmSean

Trying to stay as real as possible to the video, I’d want to know how much the container weighs when full because that would be most of interest to me if I had to move it off that chair.

So I guess I say #2 with an addendum.

6. on 23 Feb 2010 at 8:48 pmaaron

How about 3. How fast is water coming out of the hose?

7. on 24 Feb 2010 at 5:28 amSteven Peters

Sometimes the best idea is hiding in plain sight.

In my comments to the previous post I discussed question 2 (how much will tank hold) and @Aaron’s question 3 (how fast is it coming out of hose). It never even occurred to me to ask the simpler question of how long will it take the tank to fill up, partially because I could see that the video was about 8.5 minutes long. That seemed like given information to me, so I went beyond it to ask other questions.

It doesn’t have to be given to the students though. Just show a 30 second clip in the beginning and ask how long it will take to fill up. The can still do estimation and linear equations, but you don’t need to know anything about the actual container size or the water flow rate from the hose. You can just put a ruler along the height of the tank or a grid for measuring height and then they work from there.

I think either of the 2 (or 3) questions could be asked depending on the instructional goals. I do think that question number 1 has a few advantages:
(1) It requires less additional information. We already have time in the video, now we just need the height of the water in the jug.
(2) It’s a more natural question. It’s related to waiting for something slow to finish, which everyone is familiar with. I’m sure that jug has it’s capacity written on it somewhere. I’m not sure you can get the same buy-in from students on how fast the water comes out of the hose. But just show 30 seconds of that video, and it starts to get boring, and they’re already asking themselves, “Geez, how much longer is this video going to take?” It’s a more natural question.

So I think that question number 1 is the simplest, which is often-times the best place to start. I’m bummed out that I didn’t even notice it until Dan asked the question directly in this post. It’s hard to find the simplest thing sometimes.

8. on 24 Feb 2010 at 6:45 amKevin Young

My question is “Why are you giving your guests water from the hose to drink?” I’m less inclined to come to your next barbecue! But I am assuming that you would want to add some flavor to this hose water to disguise what it really is–maybe some powdered lemonade. It’s a lot of water to flavor, but how much? I would lean towards the volume question, wondering how much powdered lemonade I will need to add. Assuming it tastes good I might ask about how many guests I could accommodate on a hot day with this single tank of drink. Having said that, if I were actually trying to figure it out my inclination would be to time the filling of a 1-gallon jug and then time the filling of this tank (I don’t how to calculate volume of a hexagonal-shaped cylinder thing). In my case I would arrive at question 1 by asking question 2.

For the person who commented that the volume would be printed someplace, here’s a story of my crazy mathematician uncle (all mathematicians are crazy, but particularly if they are your uncle). At a family Christmas party I was watching fish in a large, unusually-shaped aquarium. My uncle came by and asked the volume question. I looked around and found “80 gallons” printed someplace on it. Frank pulled a small tape measure from his vest and started measuring and calculating. “I knew it!” he exclaimed moments later, “They measured from the OUTSIDE of the glass and got the wrong volume!”

9. on 24 Feb 2010 at 7:03 amFrank Noschese

I say time is more important than volume because it has more concepts wrapped up in it. However, don’t discredit the volume yet. This is a perfect opportunity for differentiation. Some students (or courses) might need the simpler volume approach. And if that covers your learning targets, great. Others might need the additional challenge of the time aspect. Or you could use the volume question as a stepping stone to the time question, for those students who *should* be able to answer the time question, but need some more scaffolding.

10. on 24 Feb 2010 at 7:30 amAndrew

I feel like I’m missing something. The volume question is easier because you would put a ruler in the picture, right? But without the ruler, time is easier than volume?

11. on 24 Feb 2010 at 9:11 pmBritt

I see benefits in both questions and I see no major flaws in either.

Trying to answer this by myself, I would present each question to either separate groups (dividing the class in 1/2) or two different classes and see which discussion proves more fruitful. That way I’m not spending time annoying myself by debating back and forth. The action research would be more accurate in answering the ‘which is better’ question anyway.

I also like the idea of two groups seeking different solutions having to share their processes along the way…

12. on 25 Feb 2010 at 9:46 amDan Meyer

My unequivocal preference is for #1: How much time will it take to fill up the tank? To begin with, as Alex and others have pointed out, each question can lead to the other. You can ask for volume and then ask for fill-up time as an enrichment activity. I can ask for time and they will necessarily need to know volume (especially if we turn the clip off before we can gather measurements for the [too-easy] proportions exercise) .

Crucial to my work with remedial students has been a style of questioning that draws insecure students into our conversation about math and gives them confidence to start flexing their imagination in new, mathematical ways.

Asking, “how much time will it take?” as an opener means students start guessing.

“Twenty minutes.”
“An hour.”
“Four-and-a-half minutes.”

I start writing those guesses on the board and attaching names to them. If I tell them I’m only going to take four more guesses, things get crazy. The students who aren’t usually confident enough to participate in these activities find an opening and jump in. This is how I bait the hook for learning.

I can’t do the same thing for “how much water will it hold?” because, as far as units of measurement go, my students are far more familiar with time than volume. Gallons are an abstract measurement to most. Time isn’t.

13. on 25 Feb 2010 at 7:18 pmA. Mercer

There is one reason why questions are not asked in this way in adopted textbooks in our state. The questions on the test are asked the same way they are in the text book. Example from California’s Sixth Grade Math release questions:

A Ferris wheel at the local fair has a diameter of 52 meters. Which expression can be used to find its circumference, C, in meters?

Rather than,

How tall is the Ferris Wheel
[with suitable illustration provided]

You Dan, are trying to prepare them for life, the text book is preparing them for the test.

14. on 28 Feb 2010 at 8:01 amDan Meyer

@Alice, let me quote anonymously from an e-mail I received this morning:

I have adopted your inquiry-driven, authentic style and worried that while they were learning, they wouldn’t pass the test. What I found was that as they moved toward critical thinking, the drill-and-kill tests actually became pretty easy for them.

Which isn’t to disagree with anything you wrote. But I have found my students very adaptable to standardized tests, if for no reason because the tests require much less rigorous thought than the stuff we do in class.

15. on 28 Feb 2010 at 11:16 pmElizabeth

Crucial to my work with remedial students has been a style of questioning that draws insecure students into our conversation about math and gives them confidence to start flexing their imagination in new, mathematical ways.

Asking, “how much time will it take?” as an opener means students start guessing.

“Twenty minutes.”
“An hour.”
“Four-and-a-half minutes.”

I start writing those guesses on the board and attaching names to them. If I tell them I’m only going to take four more guesses, things get crazy. The students who aren’t usually confident enough to participate in these activities find an opening and jump in. This is how I bait the hook for learning.

I love how you invite these students to join the conversation. In order to engage these discouraged, “remedial” learners, we need to lower the stakes of joining the discussion – not raise them.

Thank you for continuing to model this practice for those of us who are coming up to speed on your methods.

16. on 04 Mar 2010 at 3:08 pmBrendan Murphy

Why not ask, choose your favorite question, “How long will it take to fill up the tank, or What is the volume of the tank?” Your group can explain their solution to the class at the end of the period.