When they’re wrong…, or, how the heck do I scaffold?

I started this post a long time ago (maybe two or three months ago), but scrapped it. But I’ve decided to finish it up and make a little plea for advice at the end.

What do you do when you ask a question and get a totally wrong answer? Okay, this question screams newbie, but it happens to me enough and I often get caught in an awkward situation. Let me explain.

A completely made-up but not unrealisitic example:

Me: So we now we have this: x^2-2x=-1. Where do you think we go from here? What are we trying to do again? StudentX?

[Context: We’ve learned how to graph quadratics, use the quadratic formula, complete the square, factor, and seen equations like this all year. It should be second nature to them. And for many it is, but for some it isn’t. The problem is this: we’re way beyond this. We’re working on some other concept, and these gaps force me to veer away from the current lesson and take a bunch of steps back to reteach these things to the few that don’t get it…]

StudentX:  Um… well, we could add 2x to both sides…
Me: [awkward silence while I think of what to say, because I don’t want to do that…]
Me: At every step, we want to ask ourselves: (1) why do we do that? and (2) what are we trying to find out? So why would we add 2x to both sides? What are we trying to do?

[Context: Even when they are on the right track, I will ask this question. I want them to think about every move they make.]

StudentX: I don’t know.

At this point, I’ll ask what type of equation we have, and what we know about it. StudentX will finally get it (“quadratic!”), and we’ll move on.

Sometimes I don’t make it a drawn out process. If I’m in a rush, I will ask if someone else has a different idea and call on someone who I know will have the right answer, and then move from there. And then I’ll return briefly to the original idea and explain why it won’t get us to where we want to be.

But this interaction takes 3-5 minutes, I know 80% of the students in the class are bored, some are trying to whisper the answer to the student, and we get held up.[1]

Of course, I’m all about meeting students where they’re at. And I’m happy to review. But these moments happen all too often, and using every one of them as a teachable moment takes too much time and would be bad practice. I have a curriculum to cover. Taking three steps back constantly is tough.  

That tension, between moving forward in the curriculum and making sure students are up to speed on the older stuff, is palpable. 

I often feel like I sacrifice the majority of the class when I do too many of these types of things. I don’t want to praise a wrong answer (“That’s a great idea, but I’m not sure it’ll help”), I don’t want to scare a student from speaking in class (“No”), I don’t want to spend a lot of time on a basic skill that the rest of the class knows, I don’t want to make the student feel dumb or ignored (“Anyone else have a different idea?”).

I’m afraid I’ve done all three.

To make this into a truly teachable moment would require me to add 2x to both sides, and then stick with the student and ask them what next. And just stick with them until they see that they’re stuck. But I tend to only go down really wrong paths in math when we’re learning something new and we have the time to have these dead end explorations.

Basically, when it comes down to it, I recognize that I still don’t know how to organize and manage a differentiated classroom well, how to scaffold lessons, how to keep everyone engaged and learning, while still moving forward in a fast-paced curriculum. It’s not that I don’t try. About 30% of my students have some learning difference or another, and I do think about that when I’m designing my lessons. I do. But what I’m doing isn’t working. At least not as well as I’d like.

I think that in addition to classroom management, this is one of those big topics that doesn’t often get explicitly addressed in teacher blogs. Maybe that’s because a good many teachers do it without thinking about it – it’s natural. But even though I do a lot of things naturally well, planning a scaffolded lesson for a pretty differentiated class isn’t one of my fortes. Yet.

So anyway, if you know of any blog posts or websites, or have any advice, holla out in the comments.[2]

Yeah, I know, I know. Everything about this screams “Newbie.”

[1] One of my fears is that I’m going too slow for a bunch of my kids, and I’m not sure where my focus should go. The middle of the road? Those that don’t get it? Those that do? For me, I think a complicating factor is that I was always one of those kids who did get it, and really quickly. I identify with them. I don’t want those kids to be bored. And I feel guilty because I am pretty sure they are bored.

[2] Five or so years ago, I read about differentiated classrooms in one of my teaching classes, but the readings were all academic mumbo jumbo with no connection to reality. I’m looking for something useful.

My Blogroll

I’ve been meaning to put a blogroll up for a while. But the problem is that my blogroll is constantly evolving, and I wanted something that updates as I update.

Well, my RSS reader netvibeswhich I’ll tout as currently the Best. Thing. Ever. — allows you to see all the blogs I read, updated. So click on the netvibes icon on the right and check out some of the amazing blogs out there. Without further ado: my blogroll.

Bad Students Or Bad Teaching?

Okay, so here’s a story I somehow missed from a few months ago. A lecturer of writing at Dartmouth created quite a stir with a threat (then removed, then reinstated) of suing her students and some colleagues. Why? Discrimination. (Honestly, though, who knows of what kind. Even after reading everything, I’m at a loss.)

There have been a bunch of writings on the topic, most of which I’ve read, but you can get the overview with this initial article, and then chase it down with a much more interesting interview with the lecturer herself [1]. She alleges, when it comes down to it, that her students were bullies because they didn’t agree with or show sufficient respect to her and the ideas she was proffering.

The reason this controversy spoke to me is partly because the lecturer was teaching a class on Science, Technology, and Society. That’s my undergraduate major and an interest of mine in graduate school (and an amateur interest of mine today). And teaching STS to undergraduates is tough. Believe me — not only did I take a number of courses where the professors and their TAs had their hands full, but also because in graduate school, I TAed for a couple courses which were on STS or STS-themes.

It’s hard work. Here it is laid bare. Getting students to understand the concept that perhaps Science should be spelled with a lowercase ‘s’ and that it doesn’t necessarily always progress to Truth with a capital ‘T’ is mind-blowing for them. It takes a long time for them to even grasp onto that idea. That science is somehow intimately related to culture is contradicted by everything they’ve been taught. Many initially rail against these ideas because they think of science as this Objective thing which can’t have anything to do with culture. It deals, they think, with mathematical equations and physical laws of the universe, and has very little to do with the people who are writing the equations and deriving the laws of the universe

The lecturer notes in her interview:

So there was immediate friction, because basically the concepts that I was trying to bring to them were concepts I was not inventing on my own. They were concepts that were part of the field, and I was trying to bring it to the table. It offended their sensibilities, because the whole course of “Science, Technology, and Society” was about problematizing science and technology, and explaining the argument that science is not just a quest for truth, which is how we think about science normally, but being influenced by social and political values. Now I’m not telling you this to convince you of this. I’m just saying that this is the framework with which I approached the course—that I wanted to bring this view that science and technology; there’s an ethics behind it.

Once you can get them to cross over to the human side of things, you generally can then start talking about scientific innovation, paradigm shifts, and the cultural side of science. It’s hard work. And you can’t win over every student. But the best and most fruitful conversations that I had with my students as a TA, that I had as an undergraduate in my many STS classes, that I had in graduate school with other graduate students, was fighting over the framework. Disagreeing with it, playing around with it, falling in love with it. It’s controversial stuff, not the least of which is because some of it’s hard to read, some of it is hogwash, and some of it goes beyond being radical.

What reading the interview with the lecturer indicates to me, however, is that she probably just doesn’t know how to teach well. Who knows? I wasn’t there. But some choice-quote indicators:

I think that sometimes when you have some students and some instructors they mix like oil and water…

The whole integrity of the course, the whole academic integrity of the course was undermined because it never became about the students meeting my expectations, it became about me meeting their expectations. They abrogated that right. They abrogated, they turned the tables around. Bullying, aggressive, and disrespectful.

I talked about ideas that were strange, I came off as very eccentric. I can’t make things up, I can’t read their mind. So they would use any type of vulnerability. They would use this and write these horrible evaluations that hardly reflected my efforts and quality of my teaching.

I said what you did was unacceptable. They started arguing with me. I said fine. You think you know everything. You think you know everything without the knowledge base to boot, without the training, you think you have a command of all the knowledge in the world at this stage in your life, then I’m sorry, that is fascism and that is demagoguery.

That’s very arrogant because frankly, and I’m not trying to be an academic elitist, but frankly, they don’t even have a B.A. They’re freshmen.

I think a lot of professors are like, I’m the boss of the classroom and you listen to me, and that’s probably the norm. I’m a little more lenient, I’m a little more liberal, and I think this was kind of taken advantage of. I think also that many times when I was lecturing, many of the students would take over the class.

While they took over the class, the students that were questioning me would not question the student, but they would consistently question me.

She was probably the most abrasive, the most offensive, the most disruptive student. She ruined that class. She ruined it. She ruined it. That class actually had a lot of potential, there were some really bright kids there, but every time she would do a number of things that were very inappropriate… One of the things that she did, this is also really interesting, was that she would always ask me how to spell things. That was her thing. She would say how to do you spell this? How to you spell that? I mean—what am I supposed to do?—so I would tell her.

Yeah, I think professors are not immune from being questioned. I’m not saying that these scholars I’ve studied should not be questioned, but the comments I was getting on my papers were like “Oh, this thinker is like, the worst writer in the whole wide world,” or “This thinker thinks they know everything,” and I would be getting irrational things from them.

So yeah. I’m not sure she is knows how to teach. My sense is — and again, who am I to know? — she didn’t introduce the concepts slowly enough for the freshman. They’re hard concepts. She might not really have a (good) classroom management style, so the students ended up taking over the class. And the biggest thing that drives me nuts about this? She didn’t capitalize on their questioning, their dissent. That’s where the learning takes place, because it’s in the dissent that their confusion can be seen. Their assumptions teased out. And the truly academic dialogue can take place. She seemed to want her students to get it because she said so.

Academia isn’t about teaching. At least not at the big research universities, where research is privileged. That’s one of the huge reasons I had to leave grad school. The blame probably should be spread out a bit more. Still, a fascinating case study, tying together my interests in teaching and my interests in STS.

[1] A whole slew of links are here.

Student versus Teenager

I was never them. I can’t relate.

I was looking online for something related to my school, and I came across a LiveJournal by a student (from my school way before I came). He writes sporadically about his senior year (sometimes writing from class)!

Some things he says:

  • I just did absolutely awful on my bio test and I am really disappointed in myself. Weinsieder just farted and I’m in Stats… this is great. Another wonderful beginning to a day in the best school on earth….
  • Last night was pretty fun now that I think about it. Smokin a fatty and then going to Wo Hop then the Knicks game, sounds like a decent night. The problem was… marijuana is not the drug for the Knicks game. I was about to pass out right in the seat.
  • F*** YOU… I can’t believe you. I can’t believe how you can ruin me in 2 minutes of conversation. I don’t know how you can live with yourself, with just killing me repeatedly… I can’t even describe how I feel and how much I want to punch something but I’m not going to, I don’t want any more scars on my hand and I don’t want any more scars period.
  • Weird a** night. Went to the party… threw up..drank a lot… broke my knuckle…went to Vegas on Smith St… met some girls… went to their crib…
  • I was just thinking about going to Europe and how it is going to be amazing. I hope I survive and don’t go to jail and don’t get poisoned. I’m going to be rebellious and get an ear-ring before I go. My parents can’t rip my ear off if I’m in another continent. I have also been thinking about prom and how I really hope it is somewhat okay.
  • Well hmm I dont know what to say. Today I did a lot of loitering, not knowing what to do with myself…. trying to pretend to sleep. I realized that I have to start playing basketball again because 2 years ago I was pretty good and now I can be better if I get my act together. School is wwwiiiiiiiiiiinddddinnnng down and I am getting more and more aburrido.
I’m a young teacher; it wasn’t so long since I was in high school. But clearly we only see a small fraction of our students’ lives. And reading this journal reminded me exactly of that fact. Even though it often feels like they are our jobs, and even though we see them almost every week day, and even though we’re with them for a year, they don’t see us proportionally. We’re just one small piece of something much larger. We’re just one part of a constellation of teachers, while there are other constellation — many constellations. Of friends, of family, of peers, of lots of things.  I started thinking more about this recently, but after reading these journal entries, I’m expanding how I have to think about them.
Back to the kid above. I started out by saying “I was never them” — which is true. I can’t relate to many of things this kid talked about. My life was incredibly different, and I’ll freely admit that. But what comes through in these posts is more than just the drinking, partying, and apathy.
And so I lied. Because I can relate. The emotion drips through in everything he writes — the ennui, or anger, or angst. The boredom. I remember the boredom.  He cares, when he’s trying to project he doesn’t. He thinks, he feels, he is a creature whose life is a series of contingencies, he is trying to figure out who the heck he is and what he’s all about.
And let’s be realistic: we’re only a (very) small part of that.
We matter, but only a small bit. We teachers see one small slice (50 minutes of class) of something really complex (a whole life). And I don’t think I ever really knew that until this very moment.
Will this affect how I teach? I don’t know yet. Because I’m not sure I teach to mold that young person. Dan Meyer asked this exact question a few weeks ago.
In what two ways will your male teenage students spend their free time and disposable cash this weekend? How much does it matter if you don’t know?
Right now all I’m prepared to say is that: the question has a totally different meaning to me now than it did a few hours ago.
With that said, I entreat edubloggers to go out there and find one or two livejournals written by teenagers similar to those that go to your school.
Note: I kept a livejournal in college; I might do a future post on that. Might.

Algebraic Manipulation Is Overrated

An intuition question.

Look at the function below. It may surprise you that it is a constant! For any value of x, the function g will have the same value. I’m wondering, now that you know this, if you can get a sense of why it would be a constant, without (a) using your graphing calculator, or (b) taking the derivative to show that it is 0 [that is what I did, and as a side note, I have to use this on a test or homework next year].

g(x)=\frac{\sin(x)+\sin(x+a)}{\cos(x)-\cos(x+a)}

Can you find some geometric way to see that?

It took me somewhere between a half hour and an hour of playing around to get it. I can post my solution in a couple days, but right now I don’t have the energy to find a program to draw my solution [1]. But let me just tell you: it’s beautiful. You’ll be stunned when you first do it. Yeah, the calculus way tells you it is a constant, but seeing the “why” is still a mystery. The geometric way takes a bit, but whoa nellie, you won’t regret spending the time!

[1] Or maybe I should claim there is no room in the margin! (JK)

Update: I did finally write up my solution. I quickly did something I never have done before: do my work in powerpoint. It worked fine.

Update: Mr. K solved the problem in 3 minutes and found a way to show the geometric solution. Head over to his very excellent blog to see it in all it’s glory.

Update: Besides mine and Mr. Ks, a third and perhaps more elegant solution is up at 11011110.

Of the three, I think I like Mr. K’s visualization best, even though it might not be a proof in the formal sense.

“Professional Development”

Each year, my school provides each teacher with $100 of “professional development” money. I don’t know exactly why they call it that (hence the quotation marks). For things like conferences, online courses, etc., we have a really great fund to tap into. No one I have talked to has ever been denied money from that fund. This $100 is more of a mystery. You have to submit receipts for it, and it needs to be for things relating to school. I could buy school supplies, for example. Professional development? Tenuous.

And, in fact, each year there’s a book fair with tons of books for students and teachers to buy from. It’s a fundraiser for yet another something or the other. I learned that it’s tradition for teachers to never use their money during the year, and during this week in May, pick out $100 worth of books from the fair to count as their “professional development” money. This practice is so institutionalized that you don’t ever have to take out your wallet to get the books; the people running the book fair just write your name down and the total amount you’ve spent on a piece of paper and you’re done.

Streamlined, and sweet. Just the way I like it. [1]

I’m not complaining. How could I complain about this? But I do wonder why this money needs to be couched in terms of “professional development”? (No matter how broadly you look at it, my Martha Stewart books will never be professional development.) My suggestion: why not just call it a “we like you teachers and we want to give you a little pick me up” perk and be done with it?  I like my school. But for some reason, getting a $100 and being it’s told because “we like you teachers and we want to give you a little pick me up” is just so much more satisfying than “professional development.” So I’ll pretend that’s what it’s explicitly earmarked as and go along merrily.

[1] I know you’re wondering… I bought two very smart-looking hardcover Martha Stewart books (“classic” and “new” recipes), Middlesex (Jeffrey Eugenides), and The Secret History (Donna Tartt). I’ve read the Donna Tartt book before. One of the best books I’ve ever read, hands down.