This task is one possible response to this week’s check for understanding. It was a pile of fun to produce.
Real to me. My wife and I were on a beach recently and found ourselves in this math problem. This happens to every math teacher, I’m sure. We use our own product. We employ mathematical reasoning in our own lives in obvious and subtle ways. I’ve tried to discipline myself not to miss those moments, to instead write them down, photograph them, and turn them into a task where students experience the same dilemma my wife and I did.
Google Maps. The game here is to screenshot a bunch of tiles from Google Maps, align and stitch them together in Photoshop, and then fly around that large image in AfterEffects.
Use appropriate tools strategically. The sequels aren’t optional here. One sequel suggests that the cart will start moving towards you and asks “at what location will both paths take the same time?” The other asks for an even faster path than either of the two originally posed.
In both cases, I enjoyed setting up and solving the algebraic models.
But as I contemplated solving one equation and finding the minimum of another, symbolic manipulation never occurred to me. Without any teacherly presence hovering over me, nagging me to rationalize my roots, the most obvious, practical solution was Wolfram Alpha – no contest.
A teacher at a workshop pulled off a similar move this week and felt embarrassed. He said he had “cheated.” Tools like WolframAlpha require us to come up with a more modern definition of “cheating.” (And of “math” for that matter.)
The ladder of abstraction.
Referring back to the check for understanding, here are ways the original task had already been abstracted:
- the dog and the ball are represented by points; their dogness and ballness have been abstracted away,
- very little of the illustration looks like the scene it describes, for that matter; the water and sand are the same color; the image of a dog swimming after a ball has been turned into the remark “1 m/s in water,”
- points have already been named and labeled,
- important information has already been identified and given,
- auxiliary line segments have already been drawn; the segments AB and BC and DC don’t actually exist when the dog is running to fetch the ball; they have been abstracted from the context later.
My version of the task starts lower on the ladder. You see the sand and the sidewalk. You see what it looks like to walk in each. They aren’t abstracted into numerical speeds until the second act of the problem, after your class has discussed the matter. I do draw a triangle on the video, which is a kind of abstraction. I didn’t see any way around it, though.
BTW. Andrew Stadel also has a nice task involving the Pythagorean Theorem and rates.
R.G.September 27, 2012 - 5:16 am -
I have also a nice task for your next visit to the beach: Watching a sunset, can you determine the radius of the earth by raising your head (e.g. standing up) when the sun touches the horizon, and stopping the time until it does again?
Alex EckertSeptember 27, 2012 - 6:59 am -
I really like this one a lot. I was surprised at your creation of the triangle, and then read that you grappled with that too. What about a small adjustment? When you say, “Ben wants to walk straight over…” how about a small arrow that points towards the taco truck but doesn’t create the entire segment? And same for your path towards the street? Maybe half a rung down the ladder???
Chris RobinsonSeptember 27, 2012 - 7:11 am -
I just finished up using Taco Cart in a pre-algebra 7 class, but for the goal of learning rates and proportions. I’ll come back to it for the Pythagorean Theorem. Along the way, the students came up with the abstractions needed. They have never had “math class” experience with rates or proportions (which is a tragedy in its own) but knew that to check their guesses (Ben or Dan) they would need speed (rate) and distance. They didn’t need me to tell them a formula, show them how to do the math, or really anything but give them the Act 2 information (BTW: next time I am going to withhold the road distance as well). They abstracted the problem their selves. And you know what? They never complained once about doing the math, because they wanted to see if their guess was correct. Over the course of the lesson (about 7 class periods), I had students begging me to show them the answer, asking if today was the day we get to see the answer, and one students even said he was losing sleep over not knowing the answers. I’m sure the student was being a little hyperbolic, but the fact is that they were engaged. And learning math content. But most important, becoming better problem solvers.
Mark WebnerSeptember 27, 2012 - 3:29 pm -
This task reminds me a lot of an article that came out recently – “Concreteness fading” promotes transfer of mathematical knowledge.” Seems promising.
Sean WilkinsonSeptember 27, 2012 - 5:03 pm -
RG: “I have also a nice task for your next visit to the beach: Watching a sunset…”
Sometimes you don’t need any more words (or math!)
Shawn CornallySeptember 27, 2012 - 5:25 pm -
I guess I’ll be the one to point out that the string “Dan and Ben on a beach”, when translated into binary, to trinary, then to japanese, and then put through an 8-bit filter comes out to be “C-3PO and R2-D2 on Tatooine”
JenSeptember 28, 2012 - 6:00 am -
Okay, I tried not to type this here, I really, really did. But seem to be low on self-control today.
The sentence starter “Me and _____” really grates. It’s not correct. Now, I realize that you don’t have a full sentence on your first shot, but you also don’t have an implied for or by or from or any other preposition that would make you an object rather than a subject.
Ben and I would work. Or just two arrows and both names?
Okay, I know, no one likes a grammar geek. But, it’s kind of like when Khan uses terminology loosely and or incorrectly, right?
James KeySeptember 28, 2012 - 7:44 am -
@7 Jen: since you are playing the role of “grammar police,” I feel compelled to play the role of “grammar police police.” (Who will police me?)
First, a personal story. I played bridge this week on Tuesday night. At the start of the first round, I politely asked my opponent this question: “Can I please have that bidding box?” She made a comment about using “May I” as opposed to “Can I,” to which I snarkily responded “Thanks, Mom.”
I bring this up here because I think she was *wrong.* While it may be technically correct to use “may I,” to do so would be *a gross violation of social norms.* How we communicate is highly dependent on 1) whether we are speaking or writing and 2) our audience.
Example: when you see your friends, do you say, “How are you?” “I am well!” Or do you say, “What’s up?” “Nuthin!” To those who say “I am well”: enjoy your life of being regarded as pretentious by those around you! ;-)
JenSeptember 28, 2012 - 8:03 am -
“May I” is a “gross violation of social norms”? LOLOLOLOL, I guess the little kids who still play “Mother, may I _____” need to be informed of their pretentions.
Were I playing bridge with you, it would just grate on my ears; I wouldn’t say anything since you are not my under 18 yo child. However, these videos aren’t being shown at people’s game nights, they are being used at schools, for education. So I do agree with you that the audience is important. Do you not think that schools are the places to teach proper usage, be it of math terms or grammar?
So, yes, CORRECTING someone else’s grammar in the situation you describe is impolite. In an educational forum, I still felt it needed all the apologetic phrases I threw in. (However, that doesn’t mean your grammar was correct.)
Dan MeyerSeptember 28, 2012 - 8:13 am -
Someone please kill me.
Dan MeyerSeptember 28, 2012 - 8:24 am -
OMG you guys I just woke up from a nightmare where my blog turned into a grammar inquisition. No more Dorito tacos before bed ever.
JenSeptember 28, 2012 - 9:18 am -
Of course, bad taco dreams is how many students feel about math — abstract or concrete. See above comment about doing ONE problem for 7 class sessions until students just begged to be released. ;-D
Chris RobinsonSeptember 28, 2012 - 10:19 am -
In all fairness, Jen, it was a lot of problems all wrapped around the ONE question: Who gets their tacos first, Ben or Dan? I used Taco Cart to teach rates and proportions, spreading Taco Cart over those 7 classes. Don’t worry, we didn’t spend all that time “talking taco.” We did relate whatever we were talking about back to Taco Cart however, so the students would see how what they were learning about math would help them answer their question and check their guesses. It was awesome, and the students enjoyed it. As an experiment, I am going to teach a couple of learning targets beforehand, and then release a Three Act on them. In the end, I hope to compare their engagement in the two scenarios, and see which one works better with my students.
Bowen KerinsSeptember 28, 2012 - 10:32 am -
The link to your “bent path” sequel is incorrect. I’m still trying to think if there is any more natural way to introduce the bent path concept to the problem.
A related and possibly less artificial context is refraction, which you could get by filming an aquarium and looking at how the fish’s position appears different from the top and side. “Where is the fish, really?” “What’s the shortest path from my eye to the fish?” The analysis is more challenging, but if you’re already using calculus, throwing in a little trig there isn’t so bad.
As an added bonus, if you do both of these, now you can make fish taco references.
Bowen KerinsSeptember 28, 2012 - 10:34 am -
Oh, how about a video of different people all walking possible paths but maybe it freezes once some of them get on the road? It could be more illustrative than the moving dotted line path, and you might not need to draw the triangle anymore.
Dan MeyerSeptember 28, 2012 - 2:04 pm -
FWIW, my goal here isn’t verisimilitude. That’s just a nice extra if I can manage it. I’m trying to use concrete contexts to create opportunities and motivations for abstraction. The fishbowl, with its invisible lines of sight and refraction, seems like a step in another direction.
Sean WilkinsonSeptember 28, 2012 - 3:52 pm -
That’s actually what I love about this kind problem: years after this lesson, in a physics class, some student might be learning about Snell’s law and suddenly think, “wait a second – that’s just like Dan and his tacos!” And that student might even ask herself the beautifully deep question, “WHY would photons act in exactly the same way as a math teacher rushing to the concession stand?” All of a sudden, she’s got a motivation to figure out how Feynman sums over histories actually work.
But I agree with Dan: if the goal is to help students think mathematically, then we should introduce concepts using systems (like the taco cart) that students innately understand and can directly observe.
blaw0013September 28, 2012 - 7:15 pm -
Wow, no comments on Wolfram Alpha use? I say if you ever doubted, math is kill’t! #killmath
Shawn CornallySeptember 28, 2012 - 8:36 pm -
Here’s a link to a Processing sim I just wrote that allows kids to run Dan’s walk. There’s plenty of simulations of this problem available, but my goal is to always make something that students can monkey with. I’d encourage them to think about timing, how the program does slopes, how to mess with Dan’s sand vs. sidewalk speed. All that.
The code is linked at the bottom, if your students aren’t already using Processing, they should be.
Dan MeyerSeptember 30, 2012 - 4:38 am -
@blaw0013, if there’s any way I can persuade you to abandon this new posture where you know what real math and real math teaching are while most teachers are only “teaching” “math,” please let me know how I can do that.
@Shawn, thanks for the initiative there, bud. I hope you’ll help me test out of some of the assumptions I’ll lay out next week re what this problem would look like in a digital environment that reflects some of Bret Victor’s ideas in this essay.
Jonathan NicholsSeptember 30, 2012 - 1:46 pm -
I think you’ve placed the Taco Cart CCSS entries in the Split Time entry on the google doc for 3 Acts.
[Yep. Thanks. –dm]
blaw0013September 30, 2012 - 2:10 pm -
Not so certain my ways of thinking about the development of a child’s mathematical mind is particularly a new posture. It is, in fact, what I consider my doctoral studies and research agenda to be all about–so not a new posture to me. Further, it is most certainly not a new posture in the field of educational studies, and specifically maths education. Dewey wrote Child an the Curriculum in 1902, and he clearly foregrounded the learning & knowing of the child ahead of that of the adult/discipline. More recently, the same is clearly evident in the work of Piaget as well as Math Ed scholars such as Steffe, Papert, Duckworth, and Kamii.
Some folks certainly focus on the Math of the discipline, such as Vygotsky, Wenger & Lave, Boaler, and all we see focused on by CCSS and NCTM (maybe save for Gutierrez and the work she cites in her introductory chapter to the recent NCTM Equity Monograph).
Knowing of your interest in curriculum, I encourage you to read the 1997 NCTM Yearbook, especially the chapter by the IMP authors, who also (in my opinion) value the childs’ mathematics over the specific and formal Math of the discipline. (It is a curious note that you taught in one of the very first IMP classrooms at SLVHS.)
Anyways, I like your work, not only that which thinks about how to teach Math well–which will always be necessary for access to the dominant culture in this country, but especially your role in building online communities of math educators seeking professional resources and intellectual conversations.
What I think you notice (and may not like?) is that I wish our field, which to some extent seems completely unable to let go of teaching dead & useless Mathematics, failing to embrace and consider what might be a a reasonable new Math for today & tomorrow. I think Bret Victor asks for this as well. If hose of us in this field don’t work on this, I am certain Math education will go the way of Latin.
Shawn CornallySeptember 30, 2012 - 6:08 pm -
I’d love to help. I’m really getting into giving kids already written code so that they can make it better. Mostly because that’s where coding is going as a profession, and because they tend not to be able to code up interesting problems in a time frame that fits the speed of most curricula.
Let me know what I can do. I’d love to make the problem 1st person, to see if that messes with math intuition.
Also, the code uses arctangent, which is in poor form when vectors would work much better.
Jeff LaymanSeptember 30, 2012 - 8:39 pm -
This reminds me of something one of my college professors did back in the day. He subs a dog and a tennis ball transposed over a lake in lieu of you, a taco cart and sand. Either way, it’s pretty cool stuff.
R.G.September 30, 2012 - 10:43 pm -
>RG: “I have also a nice task for your next visit to the beach: >Watching a sunset…”
>Sometimes you don’t need any more words (or math!)
You are right. That’s us mathematicians. Here (http://renephotography.wordpress.com/) is a sunset I took last holidays. I swear to you I did not think about math at the time. But maybe only because lake Garda is too small. Or maybe it isn’t? Need to think about this.
Jeff LaymanOctober 1, 2012 - 9:12 pm -
wow, I have no idea how I missed this (basically) being mentioned in the previous post. Derp.
Rachel SOctober 2, 2012 - 11:57 am -
I love this. It isn’t entirely related to what we were doing in my algebra II/trig class but we had a little bit of extra time so I went ahead and showed it to them. Their level of engagement was impressive, especially for something tangential to the class. These problems are the stuff of revolution, and I’m on board.
Andrew StadelOctober 3, 2012 - 7:31 pm -
Yesterday, I used Taco Cart with my 8th grade geometry students the other day. We had just finished solving my Basketball Travel lesson the previous day so they already had a feeling for the Pythagorean Theorem and rates. Basketball travel uses a constant rate so when my students saw yours, it was a bittersweet feeling for them. They thought they were just going to repeat the process, but quickly realized your rate would change as you transitioned to the sidewalk. This information was quickly abstracted. The students got straight to the solution rather quickly, but forgot to abstract the time conversion. Many groups had answers on their whiteboards in seconds. They thought they were done and ready to watch Act 3. It took a little prompting, “so Chase, would you tell Dan that you’d meet him at the taco cart in 325 seconds?” Once we established the conversion from seconds to minutes, they still needed help abstracting the relationship between 5.4 minutes and 5 minutes and X amount of seconds. I’m calling this the 3 Act lesson that keeps on giving.
*Thanks for the nod!
Dan MeyerOctober 4, 2012 - 7:00 am -
Yeah, good press there.
Chris RobinsonOctober 4, 2012 - 8:07 am -
@ Andrew Stadel
My students and I went through the same process. I was floored that so many students thought 3.4 meant 3 minutes 40 seconds or 3 minutes 4 seconds. That is the beauty of this #3act, it can be used for so many different concepts in math, just not Pythagorean Theorem, rates and proportions.
Jon in MaineOctober 20, 2012 - 4:49 pm -
I used the taco cart problem you’ve set up here in my AP calc class. I had my students find the bearing that Ben and Dan should take to minimize the time. It was a good opportunity to work with the kids on rules of differentiation and teaching them about Wolfram Alpha. Thanks Dan! I enjoyed working getting to meet you in Limestone over the summer.