One of my tutoring students (I'll call her P) is a freshman who is taking Algebra 1 at the local catholic school. I started working with her a little over a year ago just before eighth grade. P doesn't think of herself as a "math person". Her computation is fast and her procedures are strong, but she rarely has a sense of why she is doing one process and not another. Number sense is fairly weak (though the long division algorithm or "cross-multiply" are quick) and she often flails when presented with a problem that does not match the last five that she completed. P is a hard worker and clearly wants to do well; she cheerfully tries again when her results are not successful.
Last year, in addition to helping her with her day to day work, we did a lot of estimation and problems from the Math Forum POW bank. I wanted to help train her to listen to her common sense and use it to guide her mathematical processes. P has some strong street smarts and generally a good sense about what something "ought to" be, but often has not trusted that intuition.
This past week in my ninth grade class at school, we tried one of Fawn Nguyen's http://visualpatterns.org. Although I've looked at the site many times and heard of others using them, I hadn't presented them to a class yet. I had my students look at this one:
I asked them to think about what the next pattern would look like and to sketch pattern 28. Then I had them work on creating a rule for the n-th pattern. Most of them understood what I meant and others got stuck in knowing what I mean by n-th. After some think time, they got into groups and shared their work on white boards. There were some great conversations and it was a fantastic review of algebraic concepts and reinforced the "there are lots of ways to see the same problem".
Now, back to P. She is not in my class. Her teachers mostly do not tell her that there are lots of ways to see problems. I had planned to do something fairly similar to what I had done in my class with her. But, after I showed her the image and gave her some graph paper, she just started to draw. I hadn't given her any instructions or direction. She just started to draw patterns and circle connections between them. Occasionally she narrated her thoughts, but mostly, she just saw how they interacted. "The first one fits inside the third one, but not perfectly into the second one." "The corners stay the same each time." "The squares inside keep getting bigger by one."
She also kept counting the squares of the ring rather than working to create a rule for it. It took a lot of patience, but I held myself back from interacting with her much. A couple of times I asked her to "tell me more about that". But otherwise I stayed pretty quiet. She worked on her own for almost 25 minutes. With no problem! She was noticing things, making observations and predictions, and checking to see if she was right. P did not bring any variables into her figuring. Toward the end of our time together, I asked her about what the n-th term would be like. She went right back to proportional reasoning and assumed that it would be a direct variation. Even with all of her play, once it felt "math-y" again, she lost her intuitive understanding.
I want to find ways to help P, and all of my students, to bridge their intuitive understanding and curiosity to the structure and formalism of our mathematical system.