How would you explain to a little kid what a pattern is? Katie was introduced to the concept of a pattern about a year ago (not by me) and by this point she has a fairly well-formed perception of what it is (not one that she can necessarily articulate). When Varya said that she did not know what a pattern was, Katie volunteered to explain it to her. ‘It’s like when you have red, blue, red, blue, or red, green, yellow, red, green, yellow.” Definition by example! Then I decided to take a stab at an explanation. I explained it by saying that you need to find a rule in what you are given and continue according to that rule. Katie unfortunately was not listening much to my explanation because she thought that she already knew all there is to know about patterns. But we soon found out otherwise!

After giving them a few very basic patterns, I made the following one: one yellow, one blue, one yellow, two blues, one yellow, three blues. And this is where I realized that in Katie’s perception of a pattern, repetition is a necessary element. She continued my pattern by repeating the exact same sequence of chips that I had put down. Now this certainly made a pattern, but in what was given there was not sufficient evidence for continuing the pattern in such a way. Varya, incidentally, had the right idea about how to continue the pattern, but initially messed up the implementation of it. However, when I pointed this out to her she corrected herself by putting down one red chip and four blue ones.

I then started making a triangle (actually two separate ones) with one chip in the first row, two in the next, and three in the third. Varya continued hers by putting 4 chips into the fourth row, and so on, but Katie continued with the repetition once again: one in the fourth row, then two, followed by three. I wanted to start explaining to her that she needs to look for a rule in what’s already there, but they were getting tired and so that discussion had to be put off until a future opportune moment.

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## About aofradkin

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.

I think about this *a lot* actually. The last two summers I’ve worked with K-2 kids and both times this issue has really stood out to me. From last summer:

“Almost every one of the 180 kids I encountered this summer, no matter their age or their dancing ability, were unable to identify or describe patterns outside the standard textbook context. I think they can handle more. Not only that, I think they want more.” from the post More Than Red, Blue, Red, Blue http://mathinyourfeet.blogspot.com/2012/08/more-than-red-blue-red-blue.html

And, from this summer, “It’s a wasteland out there. We’re literally wasting kids’ time on AB patterns when we could be engaging them in some truly exciting, interesting and beautiful mathematical pattern-based play, analysis and reasoning.” from the post “Beyond Linear” http://mathinyourfeet.blogspot.com/2013/06/beyond-linear.html

I’m excited to see how this develops for your little learners. It’s fun watching and listening in to your interactions.

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Sasha, as we discussed it today, you cannot say that Katie is incorrect because she gave you patterns other that you expected. On the other hand, obviously, she would learn that if we mean a repeating pattern, we will write 1, 2, 3, 1, …. BTW, remember, when discussing functions, we talked about presenting some of them (or even all discrete functions) as patterns. And when you made the squares that Katie likes so much https://aofradkin.wordpress.com/2013/08/05/colorful-tens/ , you should have mentioned created patterns (sorry for sounding like a mentor).

And Malke in her reply made a really good point that many kids think of patterns as repetitions. The problem is that instead of being a major idea throughout all the school math courses patterns are a 3 – 5 days topic in elementary and middle schools. And because of this many students are introduced to even simplest arithmetic and geometric sequences in high school instead of the first grade, when it can be a lot of fun and an object of many math games.

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I also think that maybe when introducing sequences it might be better to do ‘fill in the missing steps’ rather than ‘continue the pattern.’ For example, give them 2, 4, 6, _, _, 10, 12 rather than just 2, 4, 6 (which might have a number of potential continuations). Of course if they think of some continuation other than the ‘most natural’ one then that would be really cool too, as long as they have a good explanation for it. I would love to hear other thoughts on this!

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I am going to try and do patterns with the kids next Sunday. But from what I’ve seen already, repetition is big with them too – it’s going to be hard to break the notion that repetition is necessary. I like your idea of having them fill in missing steps – and maybe it’s better to start not with numbers but with shapes/pictures? I’ll try to find some good ones for HW and I’ll send them along to you.

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