Let me be clear, first, that Nikki Graziano’s Found Functions are beautiful, subtle invocations of math and nature. They make me happy.
But two people have forwarded Graziano’s work my way in the last 12 hours under the heading “WCYDWT?” so I’d like to point out, for whatever it’s worth, that this is significantly narrower in scope than what I’ve been proposing for the last few years. The same goes for most tweets tagged #WCYDWT, which typically link to:
Meanwhile, I am trying to:
- recreate mathematical reasoning for my students as I find it in the world around me.
- involve students in both the solution to and the formulation of meaningful questions.
- exploit my students’ intuition and prior knowledge in the solution of those questions.
I don’t have any problem using Graziano as a classroom conversation piece, but there isn’t a question here. I don’t know how to turn this interesting thing into a challenging thing.
Yes, I could go out and take a few photographs and have students model different equations also. But in the service of what higher-order question? It’s like asking “what shapes do you see here?” It isn’t worthless but it isn’t far from the bottom of Bloom’s taxonomy either.
I’m trying to get this blog feature to a place where teachers ask themselves, “what extra resources do I need to create to make this question accessible and challenging for students?” but, for the most part, teachers aren’t even asking themselves “what is the question here?” They’re applying this #WCYDWT tag to an exhilarating feeling of connection between math and the real world. Which is great, but it’s an entirely different (and entirely more difficult) task to translate that exhilaration into something a student can discover and experience for herself.
I’m frustrated. I have no idea how to make this any clearer.
32 Comments
Tom
February 7, 2010 - 4:57 pm -Not to be too much of a cheerleader but I’d look at it this way.
You’ve inspired people to look for/at math in different ways, to see math as bigger than a textbook. That’s a big step. They’re excited enough to try to connect with it, and with you, around this idea. That’s impressive, amazingly impressive.
Be frustrated but be a little bit happy.
Elissa
February 7, 2010 - 5:47 pm -I get it. I know what we’re striving for, I just don’t know how to get there on my own yet.
But using real world pictures and shapes is at least a step up from what I’ve seen in the past.
Jason Dyer
February 7, 2010 - 5:50 pm -Oddly, I think I could do something with this…
I would include everything as given, and then ask:
Find another function that does exactly the same thing.
This comes with the bonus that every student will find a different alternate function.
Jason Dyer
February 7, 2010 - 5:57 pm -Philosophical reflection: this is a WCYDWT involving looking at math and deriving more math, rather than jumping off some real world situation.
This sort of thinking is necessary for students to eventually get to the Algebra II / Pre-Calc level — where they start making abstractions about abstractions (logarithms, complex numbers, et. al.) so just thinking straight from the real world won’t cut it.
But making the original math be based an intuitive real world scenario is a handy workaround to the abstraction-of- an-abstraction-of -an-exploded-brain problem.
MatrixFrog
February 7, 2010 - 6:48 pm -I think I understand.
WCYDWT is “Here’s an interesting question. How can we solve it?” In the process of answering that question, your students will end up learning about some branch of mathematics and honing their mathematical thinking skills.
The “Weak WCYDWT Brand” is “Here is some interesting math.” But there’s no question to answer. If they’re not already excited, it won’t make them excited.
Dan Meyer
February 7, 2010 - 9:30 pm -I think that’s a valuable question, just like I thought it was valuable when Allan Bellman asked us to model the top of a Volkswagen bug with different, translated parabola.
But that isn’t asking students to apply their mathematical reasoning to the world around them. That isn’t asking students to help build the problem, to own it. That isn’t a question that can exploit a student’s intuitive sense of the world.
Like a lot of stuff, it’s good. It’s a good use of class time and a good example of higher-order thinking. But it’s different.
Dan Meyer
February 7, 2010 - 9:30 pm -@MatrixFrog, yeah that’s exactly it.
nikki graziano
February 7, 2010 - 11:36 pm -I get this question a lot, being an art student before a math student. And I’m going to assume that you and plenty others have read this, but if not, it’s certainly worth the read, or even the second read. Learning art is what taught me to appreciate math. I didn’t like it before I went to art school. And this essay puts all my feelings into much more elegant words than I ever could,
A Mathematician’s Lament (Paul Lockhart):
http://www.maa.org/devlin/LockhartsLament.pdf
“We don’t need to bend over backwards to give mathematics relevance. It has relevance in the same way that any art does: that of being a meaningful human experience.
In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide– a relief from daily life, an anodyne to the practical workaday world.”
Thanks for the post,
Nikki
Mr. K
February 8, 2010 - 5:48 am -You’ve hit on my struggle:
I can handle the technology. I can learn the design stuff. What happens for me rarely is that I get the perfect alignment of math stuff in the real world at a level that is appropriate for what I’m teaching. I have to look, it doesn’t pop out at me, and when I’m looking what I end up settling for is often second best.
This does remind me though, that it’s time for me to write that Dunning/Kruger post.
monika hardy
February 8, 2010 - 6:13 am -I think for math to have relevance for a kid (and some will only want the abstract and some will only want match-ups) is for it to be demo’d as functionality.
The graph/math has to be doing something real – in order for a kid to want to ask – how did that happen – how does that work.. think Feynman.
I think posts such as this are a brilliant way to make your message clearer Dan. I think it always helps to show the exemplars from a 4 pt rubric…
Thanks all… great way to start the morning..
Sean Morris
February 8, 2010 - 6:31 am -Dan,
I get it — a lot of us get it. I am using your stuff and moving towards finding my own. I have at least two others teachers in my department starting to bounce off your stuff as well.
Sean
Jason Dyer
February 8, 2010 - 7:12 am -Dan: But that isn’t asking students to apply their mathematical reasoning to the world around them. That isn’t asking students to help build the problem, to own it. That isn’t a question that can exploit a student’s intuitive sense of the world.
Agreed. Hence my philosophical aside above; this is the sort of thing you’d do to start leading to students making abstractions off of abstractions, stuff where relying just on an intuitive sense of the world gives an incomplete picture.
In other words, the sort of bridge stuff necessary to get students up to a Pre-Calculus level (which is essentially just Algebra II all over again, but with all the abstractions now in their full glory).
Nikki: In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited?
No, of course students aren’t interested in compound interest. They *are* interested (I know because I’ve run the lesson in the most remedial of remedial classes before) in going through used car prices and tracking depreciation rates and arguing over why certain cars lose value faster than others.
Perhaps you might like the Wall Street Journal, but it hopefully is not too earth-shattering to realize most of your 7th grade students don’t care about it yet. The “value netural” argument that is given sometimes in attacking concrete examples ignores the fact that “real life” can be only part of what makes a problem interesting. (I have written more about this issue here.)
People enjoy fantasy, and that is just what mathematics can provide— a relief from daily life, an anodyne to the practical workaday world.
I have also written about conveying abstract beauty to remeedial students, but there’s some major wishful thinking involved as far as remedial algebra students are concerned. If we are serious about communciating to all students we need to work past simply the inner beauty of math and use every technique available.
Dan Meyer
February 8, 2010 - 7:13 am -Good call. I need to do more of that. I’d like to believe I can pick out the best use of my own instructional media. And I can certainly pick out the worst. (Which is invariably, “I’ll show it to my students and we’ll talk about it.”) It would do me a lot of good to explore the shades of gray between the two extremes, though, in addition to whatever good it does for the rest of my readership.
monika hardy
February 8, 2010 - 8:08 am -agreed – i think the shades are where the masses end up.. and none of us want to be…let’s explore there…
Sue VanHattum
February 8, 2010 - 10:21 am -I hear you – this is not WCYDWT. But I like Jason’s idea. It seems to me that part of our notion of beauty (in scenery and paintings) may have something to do with regularity, and putting functions on photos might let us think about the big philosophical question of ‘what is beauty?’ in a new way.
Since we almost never (in classrooms) start with a shape and try to fit a function to it, but do need a skill like that when we’re using math to model the world, this seems like a great precalculus project.
As I watch you flesh out the notion of WCYDWT, I’m trying to figure out how I can modify it to my teaching style. I am not nearly as tech-savvy as you are, and not as visually oriented. I think eventually I’ll be able to use the principles you’re developing, and come up with my own (very different) lesson ideas.
mg
February 8, 2010 - 10:31 am -there seem to be two corners of necessary student experience here. first, engaging with the instructor in “recreating mathematical reasoning”…using cooperative examples to learn how to ask useful questions, and making visible the math already there to find solutions. but those presented scenarios, in turn giving birth to the useful questions, are still coming from the heart/experience of the teacher, even if covertly. the most valuable part of WCYDWT to me is giving students the confidence and skills to recognize within their own spherespassionsinterestsloves specific places where those useful questions can be posed.
that in mind, could one even assign a feltron-like project, but rather than students coming up with life data points to track, they’d have to come up with a useful personal question to ask and answer? they’d probably go through a few revisions, almost like a term paper, but it could be a worthwhile exploration?
Kate Nowak
February 8, 2010 - 3:09 pm -I’ve been trying to get better at this for two years. Here’s what basically has to happen to make a successful WCYDWT lesson:
1. Lighting strikes (you observe something)
2. You recognize that lightning has struck (you say “holy *&^%”)
3. You investigate by building layers of abstraction on your observation
4. You realize that that particular abstraction fits in your curriculum.
5. You strip away all those layers to a core question interesting to a 15 year old, who (I’m sorry and draw whatever conclusions you will about me or my school system) are the least interested people on the planet.
6. You rebuild the abstraction in a way that will support the questions you successfully predict they will ask
7. You make attractive keynote slides out of it
8. You extend your original abstraction to questions that they will want to pursue to enhance their understanding
Did I miss anything?
I am good at exactly none of these and I tend to lose my way around #4 and a half. I don’t know if it can be taught with any kind of reliability (how can you assign “step 1: be inspired”?) At least it’s a problem that’s worth beating my head against for a couple more years.
Anyway I know that wasn’t really the point of this post, and yes, people need to stop slapping a WCYDWT tag on any picture that kind of looks like math.
curmudgeon
February 8, 2010 - 4:46 pm -I think, for me, that the part that doesn’t ring true with those found photos (as nice as they are) is that the math has nothing to do with the situation despite the approximate match between the object and the function (or relation).
Burt
February 8, 2010 - 5:36 pm -People might think that presenting math through digital media is the essence of WCYDWT. But Dan said, “I’ll go one farther and throw out the digital media altogether.” IMO, more emphasis needs to be on this:
At least from what I understand of WCYDWT, the essence is the layered lesson. The digital media presents something real that the students can relate to, but the heart of the first layer is the big question. It does two things: (a) it draws on students’ intuition so they can answer the question without using the target math concept (b) it leads to using the target math concept to answer the question. The first layer engages the students and sets up the rest of the lesson.
What I see as the weak brand is using aspects of the first layer without the big question that sets up the layered lesson.
Jennifer Potier
February 9, 2010 - 12:38 am -I see a lovely garden, and I see a mathematical function, and I am thinking – isn’t it a lovely curve, so comfortable, flexible (possibly). Couldn’t I design that curve into a modern sofa, or musical instrument? Could seeing the curve, and knowing how to find the math relative to the curve help me to figure out how much leather I might use to cover that 3-dimensional couch (or lets make it skin- the curve could relate to the curves of the skin on the body – consider re-creating skin for a serious burn victim!). The WCYDWT offers us an infinite number of ideas – and joining it with the mathematics helps us to transfer those ideas into realities, inventions, innovations, etc.
Burt
February 9, 2010 - 1:13 am -Another way of describing the weak brand of WCTDWT is that it is flat. It presents the math up front; it lacks the layers which first invites an intuitive solution to the question, and from there builds the mathematical solution while maintaining an intuitive understanding.
Dan Meyer
February 9, 2010 - 1:24 pm -Step one is where this thing falls apart for me whenever I’ve facilitated WCYDWT as professional development.
Tell me I’m naive for drawing this back around to two presuppositions:
1. You like math. You weren’t forced into this job.
2. You use math. You’re high on your own product. This isn’t a game to you. Math has made your personal life richer, easier, or more meaningful in the last week.
Because if you like and use math, from there we’re just talking about a lot of rote technical steps to translate what’s likable and useful about math to you into a challenge for your students.
Right. My students are largely under the impression that math is a game played by strange people who live in classrooms. However much I thrill to Graziano’s work, I have to realistically admit that it will put my students even farther from a meaningful connection with math.
@Jennifer, nice. You’re describing the kind of overarching question that’s missing from Found Functions and necessary for student engagement.
Yeah, absolutely. At a certain point, I need to get your thoughts on what I’m calling the exploded view of math curriculum where we take your typical textbook application problem and explode it into layers: a) an illustration, b) mathematical notation and measurements on top of the illustration, c) sub-questions, leading to d) a large overarching question.
Textbooks present those layers as a flattened whole, which leads to all kinds of awful consequences. WCYDWT says, “separate them, build them up together with your students, use your students’ intuition as glue to hold the layers together.”
Kate Nowak
February 9, 2010 - 2:19 pm -“…from there we’re just talking about a lot of rote technical steps to translate what’s likable and useful about math to you into a challenge for your students.”
I think your use of “rote” is what’s naive. It reminds me a little of Phase 1: steal underpants, Phase 2: (silence), Phase 3: profit. I’m pretty sure Parker&Stone’s underpants gnomes were supposed to be sending up investor-types whose system for making money amounted to some vigorous hand waving…but be honest “then you just flip it and turn it into a challenge for your students” is also a little hand wave-y. It’s not something anyone was born doing and hardly anyone has been taught to do, it’s a difficult skill we have to work hard for. (And the teachers on the Internet culture doesn’t seem to promote developing difficult skills you have to work hard for – it seems to promote dashing off comments after thinking about them for 30 seconds and retweeting aphorisms that are supposed to fix everything.) Anyway, it’s certainly not rote for me. That’s all.
Dan Meyer
February 9, 2010 - 2:56 pm -Fine, fair enough. But the difficulty, for me, is like 90/10 weighted towards getting the inspiration vs. actually executing it.
Megan Golding
February 9, 2010 - 3:26 pm -Mr. K. said it right for me:
The best one I ever had? http://www.flickr.com/photos/mgolding/3160848346/
Last year, a bullet came through the canvas top of my Jeep, passed through the driver’s seat (not to worry, I wasn’t in there), and bounced off the floorboard where I found it the next day.
Now, I like math.
And I use math.
So, I brought this to my kids and asked my own (real) question – would that thing have gone through my leg if I’d been in the car at the time?
Quadratic functions, muzzle velocities of 9mm handguns, and a bunch of calculations later, we decided it likely would’ve broken skin. But, dang — I don’t run into these very often!
Dan Meyer
February 9, 2010 - 5:03 pm -The good news is that these packets aren’t like flowers or bananas. They’ll outlast us all.
If you like math and you use math and you want to play along with me, why don’t you write down the next few times math crops in your life, the next time you use it explicitly or it implicitly makes life a little more meaningful. We’ll brainstorm it out from there.
Dale Basler
February 9, 2010 - 9:17 pm -Maybe the pictures just remind people of you- like when you did this:
https://blog.mrmeyer.com/?p=849
nikki graziano
February 9, 2010 - 10:49 pm -I haven’t been following these comment responses too thoroughly, and I’m starting to think my original comment may have ended up being one of those “uh, yeah, so.. huh?” things, and that’s fine. I also don’t know anything about “the way you’re supposed to teach” (what they teach you in getting a masters for education) nor do I know what you teach, to what age group, etc. So I’m not really sure what “real” “teacherly” “education-y” (whatever adjective it is that would explain the other responses you’ve gotten) advice I can really offer, because I’m far from being a teacher, I’m a student, and I have a feeling a lot of people who read this are not.
Anyway! SO, I’m a student. So I’m hoping my perspective a little more carefully explained this time can help. Maybe it can’t, I don’t know.
So I used to dislike math, or at least be frustrated with it and never want to do it, I was lazy, it was stupid, etc. But playing with calculators in the back of class and poking different buttons to see what stupid little lines they’d make? Hell yes. Give me that. “Whatever I’ll look at the book when I have a test” kinda thing. Which I did. And I hate it now, but I at least looking back, I can understand what I was thinking, and try and also help (or at least try to) conjecture about how to fix the issue at hand.
So I’ve grown to love math as a student. Actually, no, I haven’t. (see also: http://sundayandwednesday.com/nikkigraziano/its-long-i-know/) I’ve grown to love math. As a human, I guess. I hate being a student in math because of all the crap that’s associated with it. I’m starting to realize the whole thing is pretty hopeless, it’s not really today’s teachers’ faults, at least not entirely. I know lots of teachers who try really hard and care about teaching, but ultimately our culture has just accepted one way of doing things and won’t let it go. But I feel really strongly about the way Math is taught, and even though I’m realizing there’s a bigger, cultural issue at hand, I definitely want to fight against it, however hopeless. Because how I ended up loving math is still really a mystery to me. Yeah, I had awesome teachers along the way, and they may or may not have subconsciously shaped the way I think about it, but I still saw it as just memorizing and spitting out, and I happened to be good at THAT, so I happened to “do well” in “math,” and once I didn’t have to do it anymore (art school, you know, “free spirits”) I instantly missed it. Or, well, after two years of not doing it. Not quite instantly. But I was THAT kid in the back of the class in school! Who didn’t care! I probably really pissed a lot of my teachers off, actually, now that I think about it. I was the embodiment of the problem, the unmotivated, don’t-care-attitude, sassy little twerpy kid who makes teachers say, “WHAT IS IT. WHAT am I doing WRONG? HOW can I FIX THIS KIDS BRAIN.”
Now I’m not trying to say that the way art is taught is perfect, also there’s no “do this and it works and you’re done” method. I probably didn’t even have to say that last part, but, whatever. Art school is VERY very far from perfect and I’m also not sure how to even think about “fixing” that monstrosity either, but at the very least, I know that the whole “way” “art school” works–the process of playing, discussing, making bad work and being okay with it, open critiques, defending my ideas, learning what ideas of my OWN to trust, and then actually trusting those ideas, seeing what amazing things my professors were capable of, (I’ll stop, sorry)–are the concepts that made me look at a friend’s Calc homework one day sophomore year, and say WOAH. WOAH WOAH. I GET THIS. I REMEMBER THIS. THIS IS SO SO BEAUTIFUL. There’s also the obvious difference between college and K-12, but, I don’t know. I took Calc I in high school cause I was “good at it” and I wasn’t really all that excited by it. Then two years later I fell in love with differentiation? So it is possible.
I’m going to offer some more “idea stuff” and sorry if it’s repetitive, I’m just trying to think of any initial stuff that I looked at that made me relate to math and really enjoy it.
Radiolab? It’s amazing. This might be the clincher of what convinced to go back to school for math.
http://www.radiolab.org
Again I’m not sure what age you teach, it might be too heavy for say seventh or eighth graders, probably even high school freshmen, it takes a lot of patience to just sit and listen. But if you haven’t checked it out for yourself you should. They’re good at demonstrating a sort of abstract thinking that’s necessary for math and science, but also really really beautiful, like how poets and artists and “right brain people” see little details that are almost unimportant.
Ellsworth Kelly. He’s a painter. No no sorry, he’s a “conceptual artist that paints” but since conceptual art is supposed to be the “mad scientist” of art-isms, I’m assuming kids won’t really have to know anything to like this. Here’s a link of MoMA’s below, it’s got some of his work on it, but if you find that you’re interested and poke around the internet some more you’ll definitely find them.
He looks at the 3D world around us, but sees only the 2D plane of his perspective, like how in my photos if I moved it’d be a completely different curve to map. So he makes these canvases into the geometric shapes that he sees as 2D. The titles of the works from this series usually give you a good idea of it’s supposed to be. This is a good explanation:
http://www.moma.org/collection/browse_results.php?criteria=O:AD:E:3048&page_number=13&template_id=1&sort_order=1
Hope this is of some help. Especially if you just read that entire thing and it wasn’t. :\
N
nikki graziano
February 9, 2010 - 11:13 pm -I ALMOST FORGOT THIS: This is so good.
Dan Meyer
February 9, 2010 - 11:36 pm -@Nikki, thanks for stopping back by and for offering a bit of your own backstory. Personally, I always enjoyed math but experienced something of the same headrush/epiphany you did (your second time) with calculus. The whole thing seems profound and philosophical even now.
I’m guessing the sort of educator who reads blogs and writes comments isn’t the sort of hack teacher who screwed you over in your early math education. The crowd around here (including myself) screws up, sure, but more for lack of any better ideas than for negligence or lack of ambition.
Which makes your Found Functions invaluable for those of us trying to engage the sassy, artistic twerp in our back rows. Same goes for Radiolab, which is a fan favorite around here.
My own particular obsessions have drawn me to a style of mathematical inquiry tied to digital media I dubbed What Can You Do With This? It bears enough surface similarities to your own Found Functions that some readers started forwarding them along. It was worth it to me here to draw a line in the sand distinguishing your work from mine.
At the same time, I’m glad to have a slightly larger bag-of-tricks than I did week ago. So thanks.
Alex
February 10, 2010 - 5:57 am -Hello all. I’m recommending “A Mathematician’s Lament” by Paul Lockhart for anyone who is interested in opening up others lives to math. Reading through some of Dan’s recent thoughts on WCYDWT and hearing his frustrations will make you feel as if Dan wrote the book.
My school had meetings this week to brainstorm ideas for ways to incentivize our sophomores to care about the CAHSEE. I seemed to be the only one who thought this was a joke. As if I don’t have better things to do with my time than find a way to bribe a 15 year old to give a damn about the CAHSEE.
It took me a while, Dan, but I’m starting to understand your passion for WCYDWT and I’m going to start putting more of an effort into recognizing them when I see them. Apparently I have meetings at my school where I can make good use of time to do just that.
In the meantime, go get the book!!!
Dave
February 10, 2010 - 11:28 am -The foliage equation is more general mathematical play and curiousity; it’s a solution in search of a problem. Finding the fastest line at a grocery store, figuring out how much money is in your coin jar…those are problems that you can solve the math way, or the non-math way.
And, even if the math way doesn’t give a huge benefit, it’s still an estimation lesson. Students start to get a feel for when a situation can benefit from more precision and when a quick estimation will get you 80% of the way in 20% of the time.