Is amazing how hard it is to get down to those higher order questions, those Ungoogleable investigations. Most educators appreciate taking some time out to revisit what a higher order question looks like again! :-)

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.

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.

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?

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.

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.

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.