Nathaniel Highstein engineers a counterintuitive moment about graphing, one that subverts his students’ expectations and creates intellectual need for new knowledge:

I love this problem because the answer becomes totally clear when you make a time vs. elevation graph — and the answer violates nearly everyone’s expectations and leads to a surprise! Many students got stuck in their initial guess, and even when we went over together what the intersection of the two lines implied, they tried desperately to draw a version of the graph where the two lines didn’t intersect. When they figured out that even skydiving down wouldn’t work, some resorted to teleportation.

Option 1. Explain how to use place value.
Option 2. Explain how to use place value while first asserting its usefulness to humanity.
Option 3. Explain how to use place value while first putting students in a position to experience life without any kind of place value.

Anna Weltman took option three:

Three fingers is Na Na Na. But four fingers — now, that’s a lot of fingers. Na Na Na Na is quite a mouthful and it’s getting hard to tell the numbers apart. Here is where the cavemen bring in a new word. Na Na Na Na is Ba.

We continue counting. Ba Na, Ba Na Na (giggles), Ba Na Na Na — now what? The kids think until — Ba Ba, of course!

I remember Hung-Hsi Wu’s frustration with incomplete pattern problems. Paraphrasing him: “You can’t find the next term in the sequence ‘4, 10, 16, … ,’ because it could be anything.” He’s right, of course, but you can find a first term and Chris Hunter turns that fact into an icebreaker and a robust exercise in justification. He asks students to “Extend the pattern ‘Ann, Brad, Carol, … ,’ in as many ways as you can.”

Not mathy enough for you? Remember, not all teachers will have a positive attitude towards mathematics. This is a safe icebreaker. You can always follow it up with the mathier “Extend the pattern 5, 10, 15, … , in as many ways as you can.”

The first week of Exploring the Math Twitter Blogosphere asked teachers for their favorite tasks. Lots of people mentioned Four Fours. Megan Schmidt offers us an interesting cousin to that task and a useful description of what makes it effective for students.

The students also developed some interesting strategies, like grouping pairs that totaled 16 and 25. By the end of the 30 minutes, every single student had arrived at the correct solution. I’m not sure if it was the physical manipulative or the puzzle-like feel of the task, but I was so proud of this group of kids.

My guess: it’s the puzzle-like feel. (Extra credit: What makes a math task feel puzzling?)