What comes after 3?

During a recent professional learning day, I sat in a circle with my co-workers and counted. By “counted,” I mean that we spoke number words in a standard order, not that we physically counted objects. In succession, we spoke the number word that came immediately after the one previously spoken. If anyone made a mistake we had to start over from one.

The goal was to get to 1000. It sounded like this once we got rolling: “One, two, three, one-zero, one-one, one-two, one-three, two-zero, two-one, two-two, etc.” How many times do you think we had to restart? Is counting to 1000 in this fashion a lofty goal? How many counts does it take to count to 1000 (one-zero-zero-zero)?

Maybe you have figured out by now that we were practicing counting in base four. This means that there are four unique digits (0, 1, 2, and 3). Three is the greatest single-digit number. The counts can be monitored by making composites of “four.” Every time a composite of “four” is created, it becomes a new unit. Because we cannot use “4” to symbolize this unit, we use a combination of the available digits to symbolize this composite unit. I have labeled it “10four.” A visual example may look like this:

Among our group, some had done this exercise before and some had not. We experienced various moments of uncertainty and disequilibrium. It wasn’t easy! But it was such an interesting experience. As we counted, fumbled a bit, and started over, I began to notice and interpret the way we interacted with the activity. Some individuals relied on prior knowledge of base four. Some began to notice patterns. Some needed only to hear what was spoken by the preceding individual and some relied on the cadence of the sequence beginning from one. Some were confident. Some were hesitant. Some were flustered. And most were a bit embarrassed if they were the cause for starting over.

But the starting over was the beautiful part. It was awkward at first, and then became a motivator. It was a catalyst for providing us with multiple opportunities to count. It was a reminder of the important role that counting plays in constructing a “generalized number sequence” (we can thank Dr. Steffe for the new vocabulary). We sped up as we became more familiar with the sequence and we began to apply our new “counting skill” to items in the room.

Below is a picture of chairs. Try using base four to count the total number of legs. I am providing you with the base four counting sequence.

Base four counting sequence: 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 123, 130, 131, 132,133, 200, 201, 202, 203, 210, 211, 212, 213, 220, 221, 222, 223, 230, 231, 232, 233, 300, 301, 302, 303, 310, 311, 312, 313, 320, 321, 322, 323, 330, 331, 332, 333, 1000.

I would LOVE to hear from you. How did you approach this task? What did you notice? Were you surprised?

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