STUDENT: You know that i to n power just keep repeating itself- which creates a circle, BUT THAT CIRCLE IS A UNIT CIRCLE.
Now, take that point on the complex plane that I gave you, which the angle is pi/4 or 45 degrees.
If you use Euler formula, and you put e^ipi/4= pi/square root of 2 + pi/ square root of 2 i
BUT, that is just the point on the complex plane. Why that happen knowing that I just put e^ i pi/4 ?????/ but why e to the i angle gives me the exact point on the plane???? why the base must be e??? It is just a simple plane, whyy base e to the i angle????
I hope you understand what I mean, because I do not even understand myself.
ME: I think I know what you mean.
A point on the unit circle is just (cos(theta), sin(theta)). If that unit circle is on the complex plane, we write that point as cos(theta) + i sin(theta). So far, that’s no big deal: it’s just how it is, because of what we mean by sin and cos and i.
But Euler’s formula says, that same point on the unit circle on the complex plane, cos(theta) + i sin(theta), is also e^(i theta)!!
I think you are asking, how the hell (excuse me) did e get into it?
That is a VERY good question.
STUDENT: YES THAT IS MY QUESTION- but you are not answering it??
ME: What fun would that be?
-Summer correspondence with my most inquisitive student.
I think “whyy” should be a new word.
I was just figuring out how to rotate 21 kids through 7 stations in such a way that each kid does each station with a different group. And it just dropped in to try this: 1/3 of the kids advance by 1’s. (If you’re at station 2, move to 3). 1/3 advance by 2’s (e.g. if you’re at 2, move to 4). 1/3 advance by 3’s (if you’re at 2, move to 5.) And it works! I wonder if any group reconstitutes itself along the way somewhere … and what other combinations of kids/stations this or similar patterns work for ….
Figuring out a problem yourself is better than copying someone else’s down because then it stays processed in your brain. And once it’s processed in your brain, you can’t lose it because you did it on your own.
-A student, reflecting on her first term.
Since September I’ve been dealing with two things that are forcing me to grow. First, I’ve been doing what I’ve done for a year or two, only this year, the response is different. Students are more restless, with shorter attention spans, so giving them long periods of independent work time results in little work from them and little coaching from me because I am spending all my time redirecting. So it’s back to table groups and roles and more, shorter tasks and putting things in front of them to complete instead of writing a broad direction on the board.
Second, I’ve been trying to learn how to develop fluency with some basic skills. My current experiment is in learning how to teach graphing quadratics. There’s lots of great material out there. (Check out James Tanton’s course at G’Day Math). But I thought they would be able to do it as easily as me walking them through it. The skill that common core calls “Seeing Structure in Equations” seems to take time and repetition to build. It’s something I do without realizing I’m doing it. How did I not know this? Seriously, I thought we’d do one day on graphing from vertex form and maybe one additional day to review, then on to the next standard. What’s required, at least for this group this year, is one day on where’s the vertex. One day on width. One day on finding intercepts. One day on synthesis of the above. So good: that’s what it takes. But I’m inventing as I go. It feels a lot like I am just now figuring out how to do what every other teacher already knows how to do.
The student quoted above came to see me after school to get coaching on a problem because she didn’t present any original solutions last term and wants to start 2nd term off by presenting. She announced to me as she arrived that she’s “really horrible at math.” As we worked through some background problems and re-discovered what her classmates had discovered, she lit up (“Whoa! That’s really smart!”), and as she got into the new problem she wants to present, she was really enjoying it. She told me she’s been doing what a lot of students do: waiting for other people to present and then just writing reports of their solutions (which I allow, though I will taper that off as winter turns to spring). Then out of the blue, she said what’s quoted above. I said, “Wait, I want to write that down. You just reminded me what I’m doing here.”
Twenty-six of our strongest sophomores from last year have been given to me as juniors, to take algebra 2 and pre-calculus in one year. About twenty are serious and ready to work. About six are boisterous and unable to hold their attention on listening to one person for more than about 90 seconds—but they do quality work when they work. All twenty-six would clearly (to me) be bored in our regular algebra 2 class. The knee-jerk reaction: the ones who know how to stay focused can stay, the other six have to go. But that’s the reaction that has put thousands of students, primarily low-income students of color, in courses below their ability level for decades. So the right thing to do is keep them all, and work explicitly on classroom behavior skills. And the only time that doesn’t feel good is when the first twenty students keep telling me they can’t concentrate, that the six don’t belong there, that I should get rid of them.
And there you have it: the single most influential tension in the US education system. It’s why we have private schools and charter schools and parochial schools and good schools and bad schools.
Announcing a Global Math Department webinar on Problem-Based Course Design, Tuesday, 10 September, 9pm Eastern.
Following up on a #TMC13 session, we’ll have an open discussion on ideas and questions people have about designing their own problem-based courses. All are welcome – if you didn’t catch the TMC13 session, you can catch up by looking over the materials. Questions or comments? Leave them here or #PBCmap on twitter. See you then!