A life of whining

I know, I know, teachers are always complaining.

But what’s even more terrible is that we teachers don’t tend to share our success with each other. We do have successes, I swear– even though they tend to come in small numbers and sporadically. Still, we keep them to ourselves.

So in this post, I’m going to brag.

In my calculus class yesterday, my students struggled hardcore with the chain rule. They didn’t quite get substitution for the more involved problems, and I’ve been trying to hone their intuition explicitly — saying I don’t want them to do problems formally. I want them to practice “seeing” the solutions.

Let’s level here: getting an assortment of calculus students to “see” anything is hard. They like rules, procedures, things they know will always work. Telling them to “hone their intuition” frightens them. There’s less certainty. But the reward of finally getting it is so much greater — because you can suddenly attack incredibly complicated problems.

So I waltzed into my classroom filled with students fretting about not “seeing” the solutions, and said: “Screw the homework for now. We’re going to get this!” And I gave them this sheet with 9 problems on it — I worked hard to come up with the idea the night before — specifically designed to “hone their intuition.” And all I can say is that: it worked fabulously. They were doing really complicated chain rule problems in their heads!

The rest of the class was spent capitalizing on this new understanding. At the end of class, one of my students said that calculus class is the first time she’s enjoyed math since 7th grade when she first learned to solve for x. Which means I must be doing something right.

I left glowing.

Tabula Rasa, or how I wanted to start the second quarter.

With the start of the second quarter, I vow…

to be more creative, to start a project, to put more effort into making each lesson plan clear. Basically, to bring the enthusiasm back to the classroom that I had on the first day. I made a really great first day presentation where I made a pact with the kids: they put forth their full effort and they get me at my full effort.

I told my students its the start of a new quarter. We have a blank slate to work from. Thanksgiving break just happened. I think I will repeat the sentiment on Monday: a blank slate and a new vista of territory to cover. And I hopefully will repeat that sentiment to myself too!

With the start of the second quarter, I’m (by definition) over 1/4 done with the year. Champagne all around! Seriously, though, that opened my eyes. Everyone says first year teaching is really hard, a lot of work. It was, in the beginning, but I’ve learned where I can streamline. (I can’t imagine this is true in the sciences, history, or English… but math is incredibly easy to prepare for. Smartboard makes it time consuming, but it is still easy.) The hardest part so far is becoming emotionally uninvested. And even doing that, I think, helps me out. It isn’t that I don’t care about my students, but I need to have a hard edge to show my students I mean business. So it’s going well.

Also at the end of the quarter, I had to write comments on each of my students. This too went well. What struck me about the comment writing process is how easy it is to write comments for the students who aren’t doing so hot.

Math Club

I’m co-advising MathClub with another teacher. And two weeks ago, I presented the students with the idea of continued fractions and continued exponents. After a brief introduction, I gave them a really tough problem. Honestly, I didn’t expect them to solve it the same day, but they did.

One of the problems I used to motivate the more difficult problem we worked on was:

Find x if 2=x^x^x^x^x^x^… (ad infinitum). [problem 1]

Not getting into details, it turns out that x is the square root of 2.

BUT one student, a few days later, noticed that if you say:

find x if 4=x^x^x^x^x^x^… (ad infinitum) [problem 2]

and do the exact same analysis as you do for problem 1, you find that x also equals the square root of 2. Anyone spot the contradiction? (You basically are saying 2=4.) It turns out that problem 1 has a solution, but problem 2 doesn’t.

So it turns out that if you say k=x^x^x^x^x^x^x^… (ad infinitum), there are only certain values of k which have a solution for x. And this student decided this was something he wanted to work on.

So I let him loose… I thought the right way to think about this problem is though sequences, but this student decided to work graphically. And his method is ingenious, and he got the answer.

So next MathClub he’s presenting it.

And that’s amazing. Because he came up with the problem, and he solved it!

The Vicious Cycle of Teaching

Blah. Once you get behind in one class, it throws off the entire next class. On Monday I got behind — because I spent so much time going over homework — and that meant I only had 15 minutes to present new material. Which meant the students had trouble on homework. Which mean that on Tuesday I spent so much time going over homework — and that meant I only had 15 minutes to present new material.

A vicious cycle.

That’s happened in both my Algebra II classes and I’m going to put an end to it today. I’m going to spend a lot of time before we go over the homework taking a “mulligan” (a do-over) and re-presenting the material. And then hopefully their “ah haaah!” moment will happen and they’ll say “oh I get it now!” and be able to see how to do the problems they had difficulty with on the previous night’s homework. Well, they’re 10th graders, so hard-to-read, and the actual “oh I get it now!” verbal exclamation won’t actually happen.

This problem is actually forcing me to reconsider the content of presentation. Normally I try to ease into a formula or definition — and spend more time talking about it conceptually — and less doing problems. Not that my board is all theory and no practice — but perhaps it is too much theory and less practice. I don’t want them leaving my classroom thinking the equation for a circle (x-h)^2+(y-k)^2=r^2 is a circle “just because.” I want them to understand that:

1. the circle equation is an EQUATION, just like any other one they’ve been working with. So for a line y=mx+b, all (x,y) that lie on the line are solutions. Similarly, for the circle, all (x,y) that satisfy the equation lie on the circle. It’s pretty amazing. [It’s clear to me that some of them didn’t get this… they just see a lot of letters and symbols and squares and have no idea what they’re looking at…]

2. without knowing anything, they can easily plot a few points and get an intuitive sense it’s a circle. If the x-coordinate is h, they can find the two y-coordinates easily. And if the y-coordinate is k, they can find the two x-coordinates easily. And they at least show something that COULD be a circle.

3. they should know — but not have to rederive — that this formula for a circle comes from the pythagorean theorem. It is just a new way to look at what they already know.

But I spent all this time showing these things, and then we have little time to do simple problems like:

1. If the center of a circle is at (2,3) and the radius is 7, what is the equation of the circle (simple application of formula)
2. If the center of a circle is at (1,4) and a point on the circle is (-2,5), what is the equation of the circle (slight leap in understanding needed… have to get find the radius first…)

And so when they see those on the homework, they can usually only do a problem like #1 and #2 poses more difficulty. And importantly, something like

3. If you have the points (2,3) and (1,-7) lying at the endpoints of a diameter, what is the equation for the circle?

is super hard, because there’s a few concepts that have to be brought together (have to find the center first… then get the radius…)

It’s tricky stuff, but all this time on the conceptual level leaves them in the lurch in problems like that.

The other teacher I think does things more problem-based — where he spends the majority of the class presentation time working on problems from the book/homework, so they can do the homework. I have to find out how to plan a proper lesson plan that can do both of these things.

Math is Delicious

School is now in full swing and I am getting acclimated to the bombardment of … well … everything. Meetings, questions, classes, students. And yet, I feel like I’m doing things half-assed. So far I’m still teaching to the book — almost exculsively — and I’m only able to get my lesson plans done the night before the lesson. It makes for a very nervous, day-by-day existence, like I’m precariously perched and the slightest breeze is going to knock me down. I guess with all those bombardments hurtling at me, I need to be extra careful.

I’m dog tired now, so here are some things I wanted to jot down:

0. I love teaching. I still have a lot to learn, a huge margin for improvement, but getting to pass on something you love is a pretty awesome thing. I feel that way especially about my calculus class — even if they were crazy talkative today. It’s harder to get excited about the material in the other classes, but at least for the middle school class, it’s easy to get excited about the students.

1. I still don’t know 80% of my students’ names. It’s really awful. And they’ve stopped putting up their name cards. It’s easier for me to remember the boys’s names and faces, because they tend to stay constant, while the girls tend to change their look hourly. The good news is I took their pictures so I could study them. The bad news is: I don’t have time to study the said pictures.

2. I was so busy at school today that I didn’t eat breakfast or lunch until 2:30pm.3. I had a really good weekend, not overly-filled with work, which makes me think contradictorily (a) “I can handle this whole teaching thing” and (b) “how is it that I can still go out and have a good time while all the other teachers I’ve talked to have told me that I am not going to be able to have a life my first year? I must be doing really poor work. And I know I can do better. But that would sacrifice sleep, which would sacrifice my ability to function well during classtime.”

More to come. Now I sleep.

ultimates and penultimates

Today is the last day of “freedom” — but that is true only in words and not in spirit. I am going into campus today to meet the middle schoolers during their lunch (it’s their orientation day) and then have back to back to back meetings (literally three!) with teachers about the classes we’re “co-teaching” (a term I am using because I don’t know how else to express it succinctly… we each are teaching different sections of the same class). I also have a lot that needs to get printed out for the first day: worksheets, emergency lesson plans, rosters, etc. My list has about 20 things on it already!

Yesterday, the technical penultimate day of freedom, I worked with a new-teacher-friend for 4 or 5 hours straight at a coffeeshop (and 3 hours on my own). Lots of writing and revising. What’s insane is that I was hoping to get at least the 2nd day of school’s lesson planning done, but that was wishful thinking. I did get a lot of other important tasks done.

Some lessons I’ve learned from the last few days:

1. My initial posting with regards to veering away from the textbook is just not going to happen. At this point in time, at this stage in my career, it is just too much work for me to do to still get sleep, stay on target with the other teachers, and be happy. If it was just me, and I only had one class to prep, I think it would be manageable, but having to work in parallel to another teacher makes the task more difficult.

This is not to say that I will be teaching to the textbook. Hopefully I will act as a complement to the book. My sister, the teacher, gave me some advice I think I will heed even though I told her I wouldn’t:

[I]t is your first year of teaching and your kids will learn so much even if you aren’t reinventing new crazy lessons each day and in fact they will learn more if you are leading a balanced life and are not tooooo overworked (which you will be anyway) … you will have MANY years to refine and invent and hone your lessons and ideas. Don’t teach a bad lesson on purpose, but don’t assume all text-based ideas are bad. After all, some kids really thrive on lessons based around the text; it helps reinforce their learning process. Go for one cool lesson per week. Not per week per course. Just one lesson you feel really good about once a week in one of your courses.

I can do that.

2. I am going to be the king of “beg, borrow, and steal.” When it comes to great teaching ideas, why come up with them on your own? I’ve already hit upon the genius of dy/dan multiple times, but there’s lots out there. For example, yesterday I was wondering how to teach “why is it that you get a positive number when you multiply two negative numbers?” Seriously take a minute and think about that. It’s hard to get a good real-world example for it. And the technical explanation revolves around the consistency distributive property. But online, you just punch in your question, and you get teaching solutions. Some are not really good, in my opinion, but some could work!