a/k/a A Rubric For Applied Math Curriculum
Worst
- The problem claims to represent the real-world but its illustration is only clip-art or a line drawing.
- The problem specifies the exact method of its own solution, usually in a series of substeps labeled “a, b, c, d.”
- The problem only gives information that the student will use in the solution.
Bad
The same as worst except:
- The real-world problem presents itself as it exists in the real world.
Good
The same as bad except:
- The problem reveals no information about itself – no measurements, especially – forcing the student to decide for herself what information is relevant to the solution.
- The problem doesn’t hint at its own solution method with substeps. The student can develop that solution socially, in conversation with her teacher or her classmates.
Best
The same as good except:
- The problem hangs itself on a single, concise, intuitive question, one that any student can answer, regardless of mathematical ability. The teacher solicits guesses and records them publicly, investing the students in the outcome of the exercise, and refers back to them later, perhaps introducing the concept of percent error.
- The solution to the problem isn’t read from an answer key. Instead, it’s observed by the class together in a second multimedia artifact. The class compares the answer derived from their theoretical model to this practical answer. This is scary. The class will almost certainly be wrong but the conversation about sources of error should be embraced, not feared.
For Instance:
The question: how long will it take to fill up the tank?
WCYDWT: Water Tank from Dan Meyer on Vimeo.
The answer:
WCYDWT: Water Tank + Timer from Dan Meyer on Vimeo.
19 Comments
Riley
February 25, 2010 - 11:48 am -Teaching kids to find something interesting or important, and then to define what parts of a situation are relevant to that thing, is great. Sometimes, though, when I try to lead a class based on something like this video, I find that only a few kids are participating in the discussion. Also, and maybe more importantly, I haven’t found a way to be explicit with what I’m teaching; I haven’t found a way to help kids _practice_ this kind of thought process over and over again on their own.
My stop-gap solution has been small groups. My kids have access to laptops, so I might tell them to form into groups of 3-5 kids and attack this water tank problem on their own. This gets a higher percentage of kids involved in the process, but those that don’t get into it feel their lack of understanding that much more acutely.
Would it work to have five videos like this and give them out as individual assignments? “For each video, find something interesting and then figure it out?” I’ve had very little success in keeping kids engaged with assignments like this; what am I missing?
My discomfort can be boiled down into two questions:
1: How can I involve more students in this type of investigation?
2: How can I help my kids solidify this kind of investigation as a template of thought process that could be applied to the world at large?
josh g.
February 25, 2010 - 1:41 pm -I just went back and read your last comment on the previous post. The point you made there about the huge gap between guessing units of volume vs guessing units of time for your students caught me off-guard with an obvious-in-retrospect smack upside the head. Just thought I’d suggest that you mention that in this post for those who didn’t go back and see it there.
Also worth mentioning is that this approach not only draws students in, but it reinforces real, authentic estimation and mental math skills in a way that textbooks are (inherently) horrible at.
Thanks for spelling this out clearly; I’m following your thinking here but still new enough at this stuff that having it laid out in plain view will help keep the important questions in mind when I’m stuck in the middle of making curriculum come together.
Joe
February 25, 2010 - 5:07 pm -Dan, I’m totally with you on this line of thinking but only up until a point. I’d add a further layer to your line of questioning: who cares? Who cares how fast you can fill it up? I think that unless you get to this, you end up with what Riley is talking about. Ultimately, the motivation to learn needs to be intrinsic from the student. Just making something more flashy (as much as I think technology can help here) isn’t enough. We need to also make it relevant.
Gilbert Bernstein
February 25, 2010 - 5:51 pm -Riley,
I was about to make a comment along the lines of your question #1: Namely, “What if Johnny the genius raises his hand and just blurts out the whole answer before everyone else has a chance to think about it?”
But I think there’s something crucial about Dan’s opener here that helps mitigate this problem. Even if Johnny raises his hand and says “But, we don’t have enough information!” you can still solicit guesses anyway. Johnny doesn’t have enough information to give away the solution. Maybe an ancillary benefit of the slow reveal is that you haven’t given the top students the tools or opportunity they need to short-circuit everyone else.
Riley
February 25, 2010 - 7:04 pm -In my class, Johnny’s first comment is not “we don’t have enough information” but “we need to know the times at two different water levels, and we can extrapolate from that.” Johnny and the four other kids who understand what he means right away will have a little discussion about how to best effect a solution, and the other kids will watch. I haven’t found a good way to stop this from happening in a full-class discussion. In smaller groups, Johnny’s group-mates can be put in charge of making sure he’s been rigorous, and there can be a Johnny or two in each of the groups.
I hope it’s clear that I think this approach to presenting problems is the right one; I’m just struggling with the delivery still, especially to large groups.
Dan Meyer
February 25, 2010 - 7:09 pm -Yes! Except for the “ancillary” part. It’s a fundamental benefit of the slow reveal.
Check this out:
Barry has a fourth-grade understanding of math. He struggles with “what’s five more than twelve.” He’s a great kid who does everything he can to blend invisibly into the carpet for two hours in my class. It’s sad.
The other day – I forget what exactly we were talking about – wait, right, it was this activity and I asked, “how long do you think it would take to fly non-stop around the world on an airplane, if that was possible?” I was taking guesses. Kids were all over the place, guessing 5,000 hours, that sort of thing.
Barry sensed something in the air, something reckless in the tone of the guesses, something that said to him, “Hey! Barry! We aren’t doing math right now!”
“Wait! What are we talking about?” he said, jumping into the conversation for the first time that week.
It isn’t much but it was enough leverage for me to suggest to him that we could figure it out by checking the length of some shorter flights.
I’m not saying WCYDWT is a silver bullet. I’m saying I have yet to find a technique that more effectively enfranchises the mathematically disenfranchised.
Dan Meyer
February 25, 2010 - 7:20 pm -Though, @Riley, that isn’t to discount your idea. I dig what Kate Nowak is doing right now by tossing out “data packages” and letting her students rummage through them for mathematical models. The same could be done with “multimedia packages.” I’m not sure my remedial students are self-directed enough but your mileage may vary.
This is kind of a confounding comment. Can you clarify “who cares?”
I mean, who cares that plant cells are eukaryotic? Who cares that Jupiter has 63 moons? I’m never going to Jupiter and I’m way bigger than a plant cell.
Who cares that the next number in the sequence, “4, 7, 10” is 13? The way I’m looking at it, “who cares?” isn’t that far removed from “who’s entertained?” and it’s hard for me to think of a dimmer guiding light for a teacher than that.
Zeno
February 25, 2010 - 9:29 pm -Dan, do they really expect you to teach algebra to kids who haven’t learned basic arithmetic?
rhett
February 26, 2010 - 6:02 am -Most Awesomemest:
– Same as best except the question is changed to: If the water tank was filled with beer, how many friends could you invite to your party?
Kevin Young
February 26, 2010 - 6:08 am -I love the background conversation with your neighbor. At 2:42 we see that right away he gets what is needed! Even better is 3:03 when he says “I like the way you do it man. Just like a redneck!”
Maybe we’ll just start calling you Redneck Dan!
Dan Meyer
February 26, 2010 - 6:13 am -Yes, these are my people.
Lynn Rasmussen
February 28, 2010 - 8:34 am -Go Dan! While you’re questioning reality, I hope that you’re helping your kids question reality too.
Maybe go meta? Don’t just give them lessons. Show them how you’re exploring new ways to do the lessons. Show them this post. Then ask their opinions.
With your kids who are terribly behind, maybe go meta on their math experience too. I’ve done a lot of tutoring. I find that it’s very important to explain to kids that they are behind because they were maybe too little, too interested in playing outside, and/or had poor instruction. They never added fast enough, never got their times tables down, and then got confused in the division/fractions/decimals/ratios mess. And then school just marched on.
I explain that, when you’re little, you just get confused. How when you’re older, you can step back and figure it out. How it’s not as difficult as you may think it is.
I also say, so what if you’re behind? Of course you can’t get this class right and you’re stuck in it, but all kinds of great people catch up and there’s lots of ways to learn. You can buy workbooks or take adult ed classes. Community college has remedial math classes.
Just as with the water tank, make it real, give it context, and let them find their own way to reach the answers.
You might find that with a bit of hope, even your most discouraged kids will get curious about what they can really do in class.
Dan Meyer
February 28, 2010 - 10:34 am -Good suggestion, among a lot of other good ones.
Elizabeth
February 28, 2010 - 11:11 pm -Dan – This is the kind of insight I truly treasure from you. I love your examples of how to make a question “less helpful” and consequently more interesting.
Keep going. This works.
Jinna
March 7, 2010 - 3:57 pm -OH! I have to admit with the start of the new semester I’ve been skimming through my reader but I finally get what you’ve been saying w/ this post! Thanks Dan! I can’t wait to try one of my own. =)
Alexis Piazza
November 24, 2011 - 9:40 pm -I’m curious what y’all think about this repackaging of Dan’s goods.
I’ve been trying to create a 3-act problem for Systems of Equations, and it occurred to me that I could just do this:
http://www.youtube.com/watch?v=ieavlR30LLY&feature=youtube_gdata_player
Thoughts?
Dan Meyer
November 25, 2011 - 3:04 pm -Ha. Clever. The emptying tank isn’t linear, though. So that’s kind of a drag.