One challenge that I face, however, is that my students are used to their curiosity being satiated so quickly and easily. If they want to know the answer to something, they can just Google it. On their phone. Right there.

I’m wondering – does solving the answer to “The Ticket” permit the use of a cell phone bar code scanner? I easily got “2000″ for the answer using the Amazon iPhone app. I had a feeling my students would find the answer that way too. Yup. It took about a minute before a student took out his phone and used Google Goggles.

I was in Kansas earlier this year when this problem reared its head and threatened to swallow up back-to-back workshops. I had shown this video and we were kicking a few questions around before eventually answering, “How much money is on the walls?”

In the first workshop, a participant asked, “Why did he do it?” and I talked about the \$100,000 that Hugo Boss awards Guggenheim artists for the design of an installation. A few people snickered and I realized I had just answered our next question.

In the second workshop, we were working off Guggenheim blueprints to determine how much cash was on the walls and one group seemed disengaged. I walked over and saw New York Times coverage of the installation on several screens. The headline has the answer.

Two options here:

1. Throw suspicion on Google. I asked one group to please make sure there are really \$100,000 on the walls. I mean, what if Feldman just quoted that sum to the New York Times but pocketed \$40,000 thinking, “Who can really tell the difference between 100,000 and 60,000 bills?”
2. Ask a question that’s never been asked before. The point of the Guggenheim task is to have students model the total dollars using a) the surface area of the walls, b) the surface area of a dollar bill, and c) the amount one dollar bill overlaps the next. My students found an easier way to resolve their perplexity than build that model. Power to them. So I asked them, “What would the bills look like if there were a billion of them up there?” Eventually, you ask, “What’s the most cash they could pin to the walls?” In both cases, they have to construct the same model. They’re just solving for a different unknown. For the ticket roll task (original question: “Given a ticket roll, how many tickets does it contain?”) I said, “I’m inviting my 1,000,000 friends over for a party. I’ll need a ticket roll that holds that many tickets and I’m wondering how big that’ll be. Can I store it in this room? Will I need a shed? A warehouse?”

I have a lot of faith in that second option. It extends to any kind of task. Swap the known and the unknown. Pick a number with a lot of zeros and then build a story around it.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Dave

September 4, 2012 - 10:25 am -

I always liked the maxim that if your question can be answered in a single Google search you are asking the wrong question. Once it takes two or more tabs to correlate an answer you’ve got them.

2. #### William

September 4, 2012 - 11:28 am -

“Pick a number with a lot of zeros and then build a story around it.”

That’s awesome. When we stopped calling story problems and started calling them word problems, we lost a lot. “Story” carries the connotation of narrative structure, drama, tension, and resolution. “Word” carries the connotation of… words?

It strikes me that another great resource for us math teachers is mystery writing. Conan Doyle is a master of establishing perplexity and tension. Some of his stories revolve around the “who?” perplexity, as is standard for mysteries. But others establish the “who” immediately, and perplex the reader with the “how”. The Empty House and The Speckled Band both follow Holmes as he wrestles with how a murder was committed. Doyle even occasionally hands the reader the who and the how and perplexes us with the “why”. The Blue Carbuncle and the Red Headed League are both “why” mysteries.

There’s opportunity for us to write math stories that play with the how and why questions too.

How did Felder (who couldn’t google it!) know how many bills to get from the bank teller? How did he know how much to overlap them? There are a bunch of mysteries you can find in that one story.

3. #### Bob Lochel

September 4, 2012 - 5:10 pm -

This reminds me of a heated discussion I had with a group of HS colleagues this summer. I facilitated a professional development session on the TI Nspire CX calculators, the snazzy, color, expensive behemoths. Teachers have recognized that students are now purchasing these calculators less for their utility, and more for the fact that they can save files. And that even though there is a method for shutting off the files during tests, kids are generally sneaky creatures, and find ways around it.

For me, the argument boiled down to this: if you are worried about what kids might have stored in a calculator, look up on a cell phone, or Google at home, then you are asking the wrong questions. How have we adapted lessons and assessments to ask better questions? I’m sure there were shouting matches when we abandoned computing square roots by hand (I never learned this “skill”) or stopped using trig table (I still have nightmares about doing this in high school). So why should I be worried now that a student has the Law of Sines handy? I’d rather that they can apply the formula, or adapt it to a new situation.

Back when I started teaching, I gave a challenge problem: How many zeroes are at the end of 25 factorial? When I discovered that students were using Excel to help, I moved up to 50 factorial. (Excel used to konk out and just slap zeroes at the end). Then I moved to 100! Now I have students develop the formula for n! And the discussions are so much better now.

4. #### Justin Lanier

September 4, 2012 - 6:23 pm -

An additional option—not useful in every situation—is to foster a classroom culture that recognizes that we constantly live with the choice of figuring something out for ourselves or looking to other resources for answers. It’s important not to disparage the latter option. “My students found an easier way to resolve their perplexity than build that model. Power to them.“ Absolutely. I look things up all the time as a part of my learning.

I don’t want to constantly be ramping up my questions to match the ramping up of methods that my students use in a way that makes the whole thing just a game. I don’t want an arms race, because in the end, no one “wins” school. Also, it puts me in the position of inquisitor. I’d like it if my students (eventually) bore some of the responsibility of asking the “what next?” question after their cleverness dispatches my first query.

I really want to work on talking to students about this choice between relying upon themselves and reaching out to other resources, and also to model it for them in my own learning. Being conscious about the choices we make as learners makes us better learners—more tenacious, more flexible, and more curious.

Thanks for sharing these anecdotes and your thoughts, Dan.

5. #### a different Dave

September 5, 2012 - 11:32 am -

“Googleability” highlights that students (and teachers, apparently!) should aim for more than just “finishing” problems. We need to inspire them toward learning/exploring/solving/discovering. “That’s great, you found out online that there’s \$100,000. Could you have figured that out? If you did the math, would your math be right?”

Also, the questions “What would the bills look like if there were a billion of them up there?” and “What’s the most cash they could pin to the walls?” just don’t do it for me.

Given that new knowledge about how much money was given to the artist, I would try something like “Pretend you’re the artist. You’ve been given \$100,000 to implement this money-on-the-walls idea. What do you do?” I bet it takes about 30 seconds before the entire class is desperate to know how little money they can put on the wall so they can pocket the rest. How much difference does it make if they put the bills edge to edge versus if they put an 8th of an inch between the edges? 100 pennies takes up less 2D space than a \$1-bill, right? What about the cost of staples/glue? How long will it take…and is that too long to sucker some students into volunteering to set it up, or will we need to pay people?

6. #### Jim P

September 5, 2012 - 4:25 pm -

from what I am reading in the comments, including Dan’s 2 suggestions, I am seeing these ways of approaching this:

1. ask them to confirm, and not believe what they are told (definitely important and powerful but gives me the same worry as #3)
2. think of an extension that goes further than the current situation (but maybe lose the perplexity that is suppose to be offered by the video)
3. side step the issue by attempting to promote a classroom culture that would value these moments of perplexity (this is definitely an important point as well, although I don’t think it comes as naturally to us human beings).

In any case, I wanted to offer this. There’s always the possibility of spinning it around:

you have 100000 one dollar bills to stick on the wall. what kind of building could it be? what if you stick it on both the inside and outside? use different shapes.

That way it’ll have multiple entries for students who are not as comfortable with too many composite shapes to keep track of (they can just make a simpler one). Or if you want to push them to greater(?) heights, prompt them to include shapes that you have selected.

This idea of mine is fairly similar to the #2 approach that I listed in the way that it’s an extension. But doing it backwards might be a way to interest those people that were not interested by the initial question.

7. #### James Cleveland

September 5, 2012 - 6:47 pm -

I think the key is getting them to ask such questions. 3 Acts does foster this with the perplexing first acts, but because we live in a world with information so readily at hand, the real skill comes not from having the right answers but asking the right questions.

8. #### David Wees

September 6, 2012 - 2:57 pm -

I actually don’t mind if they Google and find the answer. “Great,” I say, “You have the answer. Now how can you figure out a (mathematical) process to find this answer?”

This happened this past summer when I used the 101qs question with the pyramid of pennies. The students copied the image URL from the site (which I forwarded to them b/c it’s much easier to count from one’s own computer than from a projector 6 metres away), pasted it into Google Images reverse image look up, found the original article, which gave away the exact number of pennies. Score 1 for innovative use of tools, 0 for mathematical reasoning.

In any case, when I pointed out that the students now had an excellent way to judge whether their processes were on track, and that I was more interested in the process, the students were happy to work on their problem solving.

9. #### Rebecka Peterson

November 2, 2012 - 2:43 pm -

Dan, thanks for taking an interest in my comment. I do feel like one of the hardest things about being a teacher is developing curiosity in students; however, I also feel like that’s one of the most exciting aspects. Here’s a question (though not new at all) that kept my students very curious and, while they tried to Google it, they came up short and decided they had to come up with an answer themselves: http://www.epsilon-delta.org/2012/10/is-mathematics-invented-or-discovered.html