A colleague of mine, Warren Christensen, posted a link to this article, about some research going on at North Dakota State University. The article is titled, “Child speech experts say don’t worry if your toddler’s language regresses.” Warren linked to the article because a colleague of his, Erin Conwell, carries out the research and because his son Owen has participated research study. But something else about the article has stuck with me since reading it.
In the article, they discuss a moment in which a child says, “Daddy tumble monkey on the mat.” The errant statement is described as a causative overgeneralization, but they go on to discuss the statement in the following light:
It was an exciting time – a huge step in her daughter’s development. “It meant she was opening it up; she was ready to go and was starting to play with language on her own.”
[The child] had just moved into a whole new phase of language processing, going from a mere mimicking of the speech patterns she’d been hearing to applying “rules” of language she’d learned by listening to others to form her own word structure.
It made me wonder about what kinds of “mistakes” I should be really excited to see my students make as they are learning math or science. What kinds of mistakes would suggest that students are moving from merely regurgitating facts and procedures to applying and playing with and rules to form new ideas?
I’m really curious to hear from others about specific examples they might have seen in their teaching, in which student mistakes could be seen as a reason to celebrate their transitioning from passive mimicry to productive play with important ideas and rules.
Teach. Brian. Teach. 
Cool – I’m going to be on the lookout for these.
I read something about this a while back – in Annette Karmiloff’Smith’s book about representational redescription. Kids say “I went” (b/c they’ve memorized the phrase) and then, when they’re getting the hang of things, they say “I goed.” Then a little while later they learn the exceptions and say “I went” once again. I’ve heard others describe it as a “u-shaped development curve” and I’ve been on the lookout ever since for u-shaped curves. I gave a talk at NARST a whlie back arguing that learning progressions without u-shaped curves were probably disingenuous. But I didn’t have much data and still don’t.
A good example of u-shaped development also comes from Karmiloff-Smith’s book . There is one about children learning to balance blocks, and especially tipsy blocks (those with most of weight toward one end). I can’t remember the ages, but younger children tend to better than older kids, because younger kids only balance by “touch”. Older children are starting to develop ideas and expectation that objects balance near their center, so many older children are unable to balance the tipsy blocks. She refers to the young kids as experimentalists, and the older kids as theorists. She and Piaget gave a talk on it called, “If you want to get ahead, get a theory.”
Can’t think of any right now, but love the idea, and will also be on the lookout.
Wow. This is great.
I’m thinking about my calculus students. School has trained them (1) not to play and experiment with ideas, and (2) to feel uncomfortable when they don’t know exactly what to do next. Despite my best efforts, I can see myself falling into the habit of reinforcing this at times. And I can see the process playing out in my son’s (now second-grade) education.
Excellent question! Here is an example that came to mind. We were doing some equations and functions with a six-year-old girl, Kirk. At some point, she fell in love with the idea of variable. I overheard her explain an example to her older brother: “So this is a number, say five, and this is another number, say seven, and their sum is ten or whatever – it’s not the point…” etc. Kirk just suspended her care with arithmetic! You know how young kids can be very punctual about recently learned facts? Kirk transcended this and directed her attention to the structure underlying computations. I was very happy indeed.
This is great. I agree completely.
First one that came to mind (though I should start jotting them down when I have those conversations with students)—student tilts axes for free body diagram, but then draws vx and vy graphs with axes back to normal (splits the initial velocity down the ramp into x- and y-components and uses those on her velocity graphs). Also, the vx and vy graphs don’t match the motion because she makes the velocity constant in one direction (the forces are balanced in that direction on her axes-tilted FBD, after all).
I was so excited to talk to her about her mistakes on that problem when she came in to review her exam. I knew she was about to have a huge breakthrough in understanding about what the coordinate system meant. And it was clear that she wasn’t just mimicking an earlier problem about something going down a ramp anymore. She was thinking about whether the forces were balanced or unbalanced and had missed what it meant to tilt the coordinate system in her clumsy new playing.
I know there are more that I’ve seen just this year, but I’ll have to write them down as I remember them.