Yesterday Katie caught me playing the recent popular game 2048. When I confessed to playing a game, she looked at it and asked me whether she could have a turn. I didn’t see why not – the rules are certainly simple enough. For those of you who have managed to avoid this latest internet craze, the game consists of a 4×4 board on which you slide “tiles” with numbers. You start out with just two numbers (two 2s or a 2 and a 4) and with each move you acquire a new one. When you “slide” the board, if two identical numbers land on top of each other they are replaced by one tile with twice the number. The goal is to get the 2048 tile.

While Katie was playing I tried not to look too closely so as not to be tempted to give any hints. One thing that I did notice, however, is that she took a long time to think about each move. As soon as she finished the first game, she asked me whether she can play another one, and then later on in the day she asked me for a third. I agreed, mostly because I knew that she was not going to have much time to play during the week, but she promised me that she wasn’t going to ask for any more that day.

In the middle of her third game, Katie suddenly looked up from the ipad and exclaimed, “I get it, it’s a function!”

“What is?” I asked.

Katie: “Well 2 goes in and 4 comes out, 4 goes in and 8 comes out, 8 goes in and 16 comes out…” and she continued up to 64.

Me: “What will come out if 1 goes in?”

Katie: “Two.”

I had introduced the concept of a function to Katie a little less than a year ago and for some time after that we had fun coming up with various functions. However, we hadn’t talked about functions for a while and I’m not sure that we ever had examples of functions that involved just numbers (we certainly discussed counting functions such as “animal goes in and number of legs come out” or “a person goes in and number of children comes out.”) I was not surprised that she noticed a pattern to the numbers but I was very pleased that she tied it to the idea of a function.

To me this begs the question, why are functions introduced so late and in such an unnatural way in our schools? It is such a fundamental concept and I am convinced that most kindergarteners would understand it and have fun with it. They already encounter it all the time without realizing it (any “machine” can be thought of as a function, spells in Harry Potter are functions, our sense organs are functions…). This would save many kids from the shock of algebra, and anything that makes math (and middle school 🙂 ) less scary is a win.

I cannot agree with you more about functions. We have to have elementary teachers who understand math as a whole, not only arithmetic. And books that introduce math concepts to five-to-seven-year olds.

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Good to know!

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Wow. I couldn’t agree with you more about thinking about teaching functions earlier. It’s just awesome that your daughter has an understanding of what a function does. Please keep me posted on how this evolves. Your daughter is noticing that one thing goes in and something else comes out, and I am wondering when she will grasp that there is a rule that guides this relationship. When are children ready to grasp this concept of ratio?

I do bookmaking with students, which includes making pop-ups, lots of folding & making patterns out of cut paper. Bookmaking is all about being attentive to relationships. When I work with young students their teachers will often not believe that their students can do this work. I know that there are some cognitive boundaries I can push, and some that just won’t budge, some that I have to wait for.

If you can figure out at what age students can understand about the rules of a function, that its rule changes one number to another, and that the function defines that relationship, this would be a great thing to know. I agree what Boris said above. And it seems to me that the earlier that students can thing of math something that describes relationships of things, rather than only dong computation, the more successful students can be with math. Or maybe not. I don’t really know. I’ll be following your progress here.

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