Amazing question on Twitter yesterday.
The reason this particular question came my way is that I’m working on the next book, which will be about patterns and have the same spirit as Which One Doesn’t Belong? and How Many?
In the coming weeks, I’ll work out some of the relevant ideas in this space. I invite you to play along in the comments, on Twitter, and by using anything I share here and reporting back (If you’d rather keep your ideas, critiques and wonderings private, hit me up on the Ab0ut/Contact page).
First up, I need to share a bit of research.
Young children are able to succeed on a more sophisticated pattern activity than they are frequently encouraged to do at home or at school.
This is an important conclusion of some recent research [paywall], and it matches the kinds of findings that are common when researchers look seriously at the mathematical minds of young children.
Here’s what that claim means. There are two types of tasks that young children frequently encounter when working on patterns:
- Make my pattern
- Extend my pattern
The researchers argue that you can successfully complete these tasks without actually engaging in any kind of mathematical thinking.
Make My pattern
If you put down a pattern of alternating red and blue tiles, then ask me to make your pattern using red and blue tiles, I don’t have to notice or analyze that pattern; I just have to place a red tile next to each red one, and a blue tile next to each blue one.
Extend my pattern
Similarly, I get into the rhythm of “blue, red, blue, red” and continue the pattern without ever explicitly dealing with the repetition of “blue red” as a distinguishing feature of the pattern.
That’s the less sophisticated patterning work that children encounter at home and school. So what are the more sophisticated tasks they can be successful with?
- Make my pattern in a new medium
- Make the smallest possible version of my pattern
Make my pattern in a new medium
I make a red-blue pattern just as before, but I give you things other than red and blue tiles. I ask you to make a pattern like mine. Maybe they are different colored tiles; maybe they are also entirely different objects. Either way, you cannot just match my pattern; you have to notice the structure of the pattern and reproduce that structure with new materials.
Make the smallest possible version of my pattern
I make a tower of red and blue cubes, and I ask you to make the shortest tower you can that still has my pattern. This task also forces you to notice the structure of the pattern rather than simply matching the original.
Before I read this research article a year ago, the book’s working title was What Comes Next? and I was struggling a bit to make this question have multiple correct and meaningful answers. The current working title is What Repeats? and I’m having quite a bit of success generating conversations with images such as the following.
What patterns do you see in each image? What repeats, and how?
(A careful reader may notice that I never answered the original question. That’s because I’m still thinking about it.)
So pleased that the third in the series is taking shape, Christopher! And fantastic that this is the subject – I agree, pattern lessons are often really unchallenging and un-engaging. Making the book though seems like more of a challenge to me than the other books, to make it possible for there to be many responses about the way the patterns work. Good that you’re having success generating conversations!
Indeed, it is difficult work Simon! It has taken a lot of effort just to nail down the question, never mind to get the right prompts to go with it. Thanks for reading along and for the support!
I’m surprised that you call say that this kind of work does not count as “any kind of mathematical reasoning.” I understand this in terms of the “make my pattern” claim, but I don’t really understand it in terms of the “extend my pattern.” Isn’t falling into the rhythm of red/blue a kind of mathematical thinking — or are you saying that it’s not mathematics until you notice the unit? (And how would that impact where we see mathematics in art, music, etc.?)
Yeah, Michael, I’m happy to say that’s a stronger statement than I need to stand by. The point isn’t whether this is any kind of mathematical thinking so much as there’s deeper mathematical thinking that young children are capable of, and that they rarely get asked to do.
This is timely as I am beginning to rewrite my pattern game. It is currently an extend the pattern type game. Perhaps I can add a level for matching the pattern using different objects.
I get what Michael is saying and perhaps this is a depth of knowledge question in that we aren’t asking in depth.
The 2nd edition of Investigations dug into many of these questions, and built up to “What comes here?” instead of “What comes next?” in K, where “here” was some number of elements ahead of the last visible piece of the pattern. Kids compared different patterns, thinking about how they were alike (2 alternating colors) and different (one is red/blue, the other is green/white). Later grades focused on thinking about what color the 10th or 25th cube would be, or what number the 10th or 25th green would be, and how you know. We felt these activities focused on the mathematical power of patterns – using what you know/see to predict what isn’t known/seen. Unfortunately, the Common Core delayed work on patterns, so most of this work is no longer in the early grades of the 3rd edition.
Megan! Perhaps you are the right person to ask…. I cannot find a second edition copy of the right book, having looked after the recommendation was made to me a few weeks back. Do you have a pdf of relevant stuff, or maybe a copy you could loan out?
Yes, we can work something out. I’ll follow up via email.