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.

In my teaching physics class, students are expected to complete some content standards, which they have to either write up ahead of time and explain to me in person or prepare for writing in real-time on a board while they explain.

Some common places that students are struggling in the kinematics standards include:

• Justifying when, why, and how Vavg = (vi + vf) / 2 *
• Using graphs analytically in order to do something, not just to merely describe what is already known **
• Reasoning about the value and direction of acceleration in non-standard cases (e.g., not freefall, circular motion, etc) ***
• Providing mathematical or physical argument to justify one’s statement, not merely substituting equations ****

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* I’m thinking of re-naming the course, “Where do all these 1/2’s come from anyway?”

** Next semester, I’m going to model more of this explicitly and have them practice

*** In this problem, they have draw x vs.t, v vs t, and a vs t, for a bouncy ball. Then, in the next set of standards they have to take data to figure out a rule that describes how much energy is lost in each bounce, and then go back and adjust their drawings accordingly.

**** Lots of physics majors get by with strong algebra skills