I saw this one in class last week:
Probability of flipping 10 heads when flipping 10 coins: (1/2)^10
Probability of flipping 9 heads when flipping 10 coins: (1/2)^9
At 1:30, a student says, “if you do that probability 3 times, you ought to get that probability at least once.”
How should I have responded to this student? (I actually have the footage, which we can analyze at some point if we think that’d be fun.)
for the first question, and 
for the second questions?
This is from yesterday’s survey, which was discussed over at this post. What do you make of the responses, particularly the differences between (2a+6) in the first response, and (2x+49) in the second?
This post is part of a series analyzing a bunch of survey results. For previous posts, go here and here.
Noteworthy:
- The kids have a ton of confidence, even in the stuff that they haven’t formally studied in class yet. (For this survey, Questions 1-3 had been covered formally, and Questions 4-5 had not.) To my mind, this continues to reaffirm that the most annoying mistakes aren’t the distortion of instruction; they’re the failure of instruction to override preconceptions.
- Kids like to say that
, and teachers like to say that this is due to overuse of the Distributive Property. That might be true, but those teachers also have to recognize that kids said that
with almost the same verve and frequency. It’s hard to blame exponents or notation for that mistake, right? So where does this intuition come from?
- A couple of kids included a
term in Q4 and a
term in Q5. I find this interesting, but I’m not exactly sure what its significance is. Is the temptation to add
and
The idea that kids walk into our classes with these intuitions is, I think, counter to the way that most math teachers talk and think about these mistakes. I think that realizing that these mistakes are the result of deep intuitions about how math should be is important. I also think thinking about where these intuitions come from is important, because maybe we can avoid setting them in earlier years.
I hope that some of you will give this survey to your students who haven’t yet received instruction on how to multiply polynomials. The original survey can be found here.
You’ll disagree with me in the comments, right? I’m counting on you all…
Thanks to Tina, we’ve got this great example of a tiny little error that crops up during complex numbers. Here’s my take on it: there’s no way that this kid would make this mistake if their problem was just “Simplify the square root of negative 4.” When the skill is laid forth is such a direct way, it’s very clear what the student is supposed to do. But when the skill is embedded in a much more complex problem, the student “handled” the negative root by realizing that this was a context that deserved a complex number. Happy and satisfied that they noticed and “handled” every aspect of the problem, the student moved on.
I like calling these sorts of mistakes “local maxima” mistakes, and I think they’re fairly common. To me, the importance of these sorts of mistakes is that they reveal the problem with testing any skill in isolation of others. I’m <i>absolutely sure</i> that this student could simplify the square root of negative four if plainly asked to. But that didn’t mean that this student was able to use that skill in this context, when there are many more things to juggle.
To me, this means that you can’t really assess any individual skill in that sort of isolation. Instead, I’d prefer an assessment system that gives students a bunch of chances to use a skill — unprimed — in the context of a fairly difficult problem. If the student can simplify negative radicals in 3-4 more involved problems, then I’m pretty confident that this kid has that skill down.
Categories
Parallelogram Problems
This is a straight up student-empathy question: what was this kid’s thought process like?
Thanks to Tina for the submission!
Editor’s Note: I categorized this as Grade 7 – Geometry in the CCSS, but I’m not really sure if that’s right. Where does this belong?
Categories
Three a week, not five a week
Hey all,
This is just a note to make something official that I’ve been doing unofficially for a few weeks.
I’m now posting 3 mistakes a week instead of 5. I think that’s more keeping with people’s ability to process this stuff and comment intelligently. Bonus: it gives me a bit more time, a bit less pressure.
Think this is the wrong move? Let it rip in the comments.
Thanks all for all your help and support and things.
-Michael