Two Worksheets

ONE

On Thursday, I’m going to be introducing absolute value inequalities. Last year I used the picture below as motivation.

I then tried to work backwards to show kids absolute value inequalities. It wasn’t too hot a success. Certainly the “application” wasn’t a motivator, and working backwards just confused things.

This year, I’ve decided to start with a warm up. Without them knowing anything, I’m going to ask them to do this for the first 7 minutes of class with their partners.

I already can see the great questioning and discussion that this simple worksheet will generate between partners. And then, when we come together: WHAM! powerful! It’s a simple thing, but Oh! So! Delicious!

After that, after we see some patterns and make some conclusions… then, then I can throw up the picture of the bag, and talk about it meaningfully. And have kids work backwards from their own conclusions to finding a way to express that region mathematically, using absolute value inequalities.

TWO

I’m introducing limits tomorrow. I pretty much have carte blanche in what I do. Last year what I did was sad. Like SAD. Like: “Here’s what a limit is. Get it?” This year, I’m stealing pretty much from CalcDave wholesale. Here’s his calculus questionairre. And here’s what I made.

Pretty much the same thing. Then I’d like to somehow have them start thinking about how to get velocity from a position versus time graph. Haven’t quite figured that out yet. Either that, or Zeno.

A great Multivariable Calculus problem

Today I gave my multivariable calculus class a problem — a problem I give every year, that I found… somewhere. Maybe MIT, maybe an Exeter problem set, maybe a textbook. And if you ever want to see kids work together, and do some good problem solving, this is a prime problem for that.

Up to now, we’ve been working on vectors — and they learned vector basics (read: dot product and cross product). Here’s the question.

You have any tetrahedron. Sticking outwards from each face, orthogonal to each face, is a vector with magnitude equal to the area of face it is sticking out of. Prove that if you add these four vectors together, they sum to the zero vector.

It’s such a beautiful problem. I don’t have a totally geometric way to explain why this is true (we do lots of good vector algebra), but I do enjoy watching everything all come together. To me, it almost works like magic.

I then had a student ask an amazing extension question. (If they’re asking extension questions, you know it’s a rich problem.) He said: “Will this always work for any polyhedra? What about figures involving faces which aren’t triangles?” I, of course, decided I loved the problem. And I desperately want him to work on this for his final project.

I love the idea of this student taking this problem and seeing how far he can run with it. I mean: hello, gluing tetrahedrons together! (I expect him to make some stick models, if he does it!)

Academic Integrity as part of Personal Integrity

As you may know, I teach at an independent school in Brooklyn. It is a K-12 school, with about 80 kids per grade in the Upper School (grades 9-12). I love that even though there are around 320+ kids in the Upper School, there is a sense of community and belonging. For this to happen, a lot of stuff has to be going right. I have my gripes, believe you me, but when you take a step back and see all the good, well, I am proud.

At the same time, our kids are under a lot of pressure. Pressure they put on themselves, pressure we put on them, and pressure they get from their parents. They tend to overextend themselves — from sports and music and dance to SAT tutors to clubs to horseback riding to writing computer programs. And that’s on top of  the regular school work we assign — which can end up being a lot.

And in our school, students cheat. (That’s probably not special to my school.)

I know students cheat. I’ve been on the Student-Faculty Judiciary Committee (read: disciplinary committee) for the past two years. I’ve seen it all. The most common form of cheating? The one that is rampant? Homework copying.

Anyhoo–I’m helping lead a group of 15 teachers to investigate “integrity” in the Upper School. Where and how we teach it formally and informally? Where we fall short? How other schools teach/model integrity? So that we can find concrete and specific ways to improve our own practice.

It’s not a bunch of fluff. One of the most eye-opening things for me was a discussion that I had with my homeroom last year about cheating. We had a frank, honest, open conversation — where I posed some questions and hypothesticals and just listened to them talk. It was a judgment-free zone. What impressed me about the conversation was that students had passionate views about the subject, about the role teachers play (read: busywork versus meaningful assignments), and their ability to voice their views confidently. What surprised me about the conversation was that the word “integrity” (or “morality”) did not enter into the discussion until I mentioned it at the end. Mostly their views were: it’s okay to cheat on small things, like homework, because you’re just hurting yourself. [1] It doesn’t really matter.

Most the other homeroom teachers I talked to told me their kids said similar things.

We want our kids to behave with integrity. We can’t force them to buy into it. But I see two things that we can do to promote personal integrity.

1. Model personal integrity as a teacher. How? Easy. “If you say you’re going to do something, do it.” [1]

2. Talk explicitly about integrity, and not just once. If it isn’t just you as a teacher doing this, but a whole slew of teachers, well, at the very least, the words and the ideas will be floating around in the school zeitgeist. As our learning specialist said: Naming things has power. Don’t just be implicit. Be explicit.

I try to do (1) all the time. Hello, my motto: clear, consistent, and fair. This year, however, I’m talking about (2) explicitly.

Specifically, one of the things kids have trouble with at my school is with collaboration. Different teachers have different expectations regarding collaboration. And what’s okay in one class is not okay in another. It can be confusing. And you know what? It’s hard to decide where the line is — between acceptable and unacceptable collaboration. There are grey areas. And saying “you can work together but you have to write up your work separately” is a cop out. The ambiguity present in that phrase is ridic.

So what I did was write three skits to be performed with another teacher, and had kids think about the following four questions:

  1. Where in the collaboration does the learning taking place?
  2. What are the positive pieces to the collaboration?
  3. What are the negative pieces to the collaboration?
  4. If you were in this collaboration, what are ways you could make the collaboration more productive?

The skits are below. [2]

I had kids take notes during the skits, after seeing these four questions, and then we had an informal conversation. It was one of the best things I’ve done in the classroom. Everyone had something to say. They responded to each other. They were thoughtful and mature. And most importantly for me, in each skit, they really thought about who was learning and who wasn’t in each collaboration. What I’m hoping is that this translates — even for one or two of them — into thinking about the way they collaborate with others on their assignments, and if there is a better way to collaborate.

I am going to have my kids engage in groupwork this year, and I hope I remember to ask them at the end of a class with a lot of group work: what was positive about their groupwork, what was negative, who was learning and how do you know, and how they could have made their groupwork better?

The big conclusion that I shared with them — and it’s something I only stumbled upon this year when thinking about academic integrity and collaboration:

I’m also going to start putting honor statements on all my assessments.

Again, not because I think it will “force” kids to act with integrity. You can’t force that. But it puts the idea of integrity out there in the zeitgeist. And who knows, if they see and hear and talk about it enough, it might reframe what students think about when they think about cheating: cheating is bad because I can get in trouble vs. cheating is bad because it diminishes the trust that others can put in me.

[1] I had this idea a while ago, but it was echoed by a teacher at my school at a recent meeting.

[2] In case you were wondering, Skit 1 borders on being unacceptable collaboration. Skit 2 is unacceptable collaboration. I made a blanket rule, after talking about this with my kids, that no student is ever allowed to give their homework to someone else, under any circumstances. Skit 3 is ideal.

Randomness, because it’s Still Early

I don’t know. I thought I was doing okay — dealt with a SmartBoard crisis, made good progress into the content of each of my classes. Yes, I’m only three school days in, and I’m rushing forward, like the train that I always think I’m on. And I was thinking “wow, I’m getting back in the groove of things.” However, I read Mythagon’s most recent post, and I’m more certain than ever that: I’m not.

I’m not unhappy. And I’m not all pie-in-the-sky. But I did have one big goal for this year, and that involved having a lot more group work and math-talk in the classroom. So far, we haven’t had that, not in any significant way. And it’s so important early on — so you can build the norms and show “hey, this is something we do.”. I hope I can get something group-work-y happening soon.

Here are some other random things from the start of the year:

1. When kids want to go to the bathroom, they just have to go like:

Then I’ll see it, and just gesture for them to go, without interrupting the rest of the class/lesson. (One of the middle school teachers mentioned this to me, and I think she said she got it from the Lemov book.)

Funny true story. When I introduced this to one of my classes, a student said: “Do I need to put one finger up for a number one and two fingers up for a number two?” I love kids.

2. I designed this poster for my Multivariable Calculus room. (It’s poster size.)

Each time we encounter a new Greek letter, I’m going to have a kid highlight the letter we’ve learned. We’ll see how many letters we’ve picked off by the end of the year.

3. The teacher I’m teaching Algebra II with this year is new to my school. In her old school, she renamed homework “home enjoyment.” It’s silly and corny, but at least it gets you thinking about homework as something other than work. (Too many negative connotations.) Last year, I tried going with the uber-dry phrase “daily practice.” Just to remind my kids what they were working on. I couldn’t stick with it. This year, I’m going to stick with home enjoyment. In all my classes.

You can see I’m already on top of it. I’m doing it on my “course conference” (where I post the home enjoyment and smartboards each day.)

4. I decided that I need to try to make sure I have at least one humorous thing in my smartboards for my Calculus and Algebra II class every day. I had this whole schtick setup on my smartboard…

… but then I decided I would print out a color picture of beebs, and write out the conversation for two histrionic students to act out in a script (while holding the picture of beebs over one of their faces).

5. I’m working SUPER hard to learn my kids’ names. But — if any of you know me in real life — you know this is perhaps my biggest challenge. I can barely get their names when they are in assigned seats and have name tents propped up. But I think I’m getting the hang of them. The worst is when I see one out of their natural habitat. Meaning: out of their seat. They might come into the math office. They might see me in the hallway. And they’ll start talking to me. And my brain just goes into overload. It’s embarrassing.

Luckily, I preempt all awkwardness around this by telling my kids this on their first day and joking about it. They know I’m bad, and they see me trying (and often failing) in class to come up with their names.

6. My non-teaching plate is full this year. I’m taking over the Upper School’s (read: high school) Student-Faculty Judiciary Committee. I’ve been a faculty representative on it for two year, and now I’m beginning my two year tenure as the “faculty advisor” (read: grand poombah). It requires a dauntingly scary amount of work. But in my opinion, it’s the best thing about my school. Which is why I’m willing to be dedicated to it.

I’m also helping create a Peer Observation Group at my school. I’m hoping it turns into a semi-formal-yet-still-informal, positive way for teachers to start migrating into other classrooms — like little bees — and collecting and cross-germinating all sorts of wonderful idea-pollen. My current feelings: I’m super excited to go look at other classrooms, while I am super anxious about anyone coming to see mine.

As well as helping advise the math club. And running the New York Math League and AMC.

More things will probably be added to my plate as the year goes on.

UPDATE: I forgot one more thing that worked well at the start of this year that I want to be sure to do next year. My co-adviser and I emailed our homeroom two days before the start of school, promising them a pancake breakfast if they all brought the locks for their lockers on the first day of school. Usually getting 11 kids to bring in their locker locks takes a week or longer. And involves a lot of chasing kids down individually. But this year, it was awesome. THEY ALL BROUGHT THEIR LOCKS.

First Day of 2010/11: Introducing SBG

For those of you who have secretly been giving dirty looks to me, because you’ve been in school forever, and I’ve been on summer vacation forever: hey, I’d do the same. But lucky for me, those looks can stop, because we officially have started classes. Today. Of course we have tomorrow (Thursday) off because of Rosh Hashana. But then we have classes again on Friday.

I’d like to talk about how I introduced the new grading system in calculus. Basically, the answer is: I took a little from each of you. Not only in the development of the system, but how I talked about it. We had a meaningful back and forth a few times, where I asked them some key questions, and got them to reflect about the system.

Here’s the general setup – in words. I’ll include my slides at the end if you want to go through them.

Students come to the classroom, see the seating chart projected on the smartboard, and sit down. They have at their desks large index cards which they make a little “name tent” so I can learn their names. I talk with them as they arrive, and I get them in a boisterous and joking mood. I’m already in a giddy mood anyway, so this isn’t hard. I’ve taught a bunch of them before, and I know others from this thing or that.

Then we go into the course. I talk about historically the problems students have had with calculus — from all teachers I’ve talked with. The big thing separating kids from calculus is a giant mountain of ALGEBRAIC SKILLS. I talk to them about how we’ll work around this by doing algebra bootcamps for the first semester. (I’ve written about this before, I hope, right? I can’t seem to find it in the archives… hm…) I talk about how calculus is actually quite simple — just a few basic conceptual ideas — but the thing that bogs students down is not being super solid with the algebraic undergirdings. So we’ll just get the relevant algebraic skills out of the way beforehand so we can focus on the calculus in each unit.

I then polish off a few logistical things (e.g. no eating and drinking in the classroom, since it is a designated lab classroom), and then I intentionally lie to my kids.

Here’s where the magic comes in, I believe. I have to lie to my kids, for the new grading system to make sense. I had to raise their anxiety about the course, to mimic the anxiety they’ve had for all their other courses. I want to play on their emotions. It’s a little cruel, I know.

So I made up a fake grading system: 90% assessments, 10% homework. 3 tests per quarter, so each test is worth 30% of their final grade. I make it a point to tell everyone that there are no retests.

I talk about how “THIS IS CALCULUS” and “IT’S SERIOUS BUSINESS” and “IT’S HIGH STAKES.” I also talk about how I don’t want to make it high stakes because I’m a meanie, but rather, I feel obligated to get them prepared for college.

I actually am super convincing, if I do say so myself.

I really emphasize that this is a do or die class, but I do so with some humor. They all laugh at the right places (see the slides below for full effect), but I can tell some of them are freaking out.

I tell them the date of their first assessment is in two weeks.

And then I ask for thoughts.

(Crickets.)

Then, when the pause is pregnant enough, when the tension is just about at its height, I say “JUST KIDDING.”

They don’t know what I mean, so I explain to them that everything I told them was a lie. I tell them that we’re not going to be grading that way, and that I just wanted to make a point — which they’ll see later. Then I asked Shawn Cornally’s question:

“What do you do if you bomb a small quiz?”

I also got some priceless, heart-breaking, answers, like Shawn did:

“Crumple it up and shove it in my bag, hoping to never see it again”
“Shrug.”
“Forget about it.”
“Mope.”

Of course, I got some non-depressing answers too, like “Meet with the teacher” and “Go over the material again at home.” Some kids know what they think I want to hear — so these responses could be that. But I know some of these kids, and some genuinely have a lot of these learning skills down already — about being proactive and on top of their learning continuously.

So I talk about how them not learning material they missed hurts me, because I am a math lover, and I want them to know everything! And I want them to have the opportunities to learn things and be recognized for that.

Then I introduce the grading system. Pretty much how I outlined in the handout. I explained how it worked, I explained the wonders of it, I explained how it’s about them learning how to learn effectively, not just getting a zillion second chances.

I talk about my own concerns with it: the amount of work it’s going to be for me, the fact that I’ve never done it before. And then I get them to talk about their thoughts about the system. What they like about it.  That goes easily. We also talk about the meaning of grades.

Then I ask them: what might make this system hard for you.

They come up with some great things: retention, grades going down, no classroom engagement grade to “buffer” their grades, etc.

I then introduce the idea of how their quarter grade is calculated: it’s all skills. And how homework isn’t included in their grades (though it is required). I ask them again: what difficulties or situations do you think might arise with this?

They again are really thoughtful and realistic in their responses: all variations on “I might not do homework.”

I then pace around the room, saying histrionically “I’ve told you my big concern for me — the work involved. Now I’m going to tell you my concern for you…” I lay out a story about it being 10 or 11 at night, and they have two or three unfinished assignments. They decide to forego the homework in calculus because “well, Mr. Shah might give me a disappointed look, but he isn’t grading me on homework.” And how there is a chain effect, with how things can quickly build up. Procrastinators beware. And how I’m here to help, but “WITH GREAT FREEDOM COMES GREAT RESPONSIBILITY.” Yadda yadda yadda.

Three comments, of quite a few, that were said by students that stood out for me, were:

“Mr. Shah, I want you to know: this is the best first day presentation yet!”

“Wow, this just makes so much sense.”

“Did you come up with this idea on your own?” (No.) “Where the heck did you find this thing?” [I said on math teacher blogs. I saw some snickering.]

Then I pretty much dove into, verbatim, David Cox’s spiel about knowledge, community, and sharing.  I even stole the pictures.  And this quotation from his blog:

I love my kids, but I also love the fact that I think this form of grading makes sense to them, philosophically and emotionally. I don’t know what will happen as this gets implemented. But I suspect I have gotten the buy in because of this first day presentation.

Full presentation here:

My SBG system

I’ve now introduced my Standards Based Grading system to the following people:

1.) My chemistry teacher friend, whose opinion I trust and respect the most on all matters pedagogical. (This isn’t hyperbolic. She beats all y’all, blog and twitter friends!)

2.) The learning specialist at my school (I had to give her a super brief and inarticulate overview)

3.) The Upper School head of school (read: principal)

4.) My department head.

And I will soon be talking to the senior class dean.

Why? I’m not about to embark on something so different, in terms of how things are done at my school, without ensuring the support of those people who will, or might be affected, by this change. Also, I was just darn excited about it and wanted to share with them what I have decided upon.

What was really heartening was that people really seemed to understand it, and interested to see how it actually panned out on the ground. As you know, I’ve been thinking and reading — from y’alls blogs — about SBG for a long time. And it took me a while to finally “see” it [1]. But because of that struggle, and thinking through all the drawbacks of SBG (logistical and pedagogical), I was able to deftly and confidently field all the questions I was asked. And more than anything, I was really heartened by the serious interest and great questions coming from the Upper School head. The last few days have made me really proud to work at my school.

So without further ado, I am posting what is my final version of the introduction to the new calculus grading system.

There it is.

Some decisions I had to struggle with to make:

1. Rubric goes from 1 to 4, not 1 to 5 if a student attempts a problem. Because as @jlanier said on twitter, you don’t want to give yourself a middle option so you can straddle the fence.

2. I decided to not include homework as part of the grade. I’m scared of students not doing it, but part of teaching them involves them seeing for themselves that doing homework [read: practicing problems] is a necessary skill to be fully confident with (and to retain older) skills. I am, however, still requiring students to do homework, and they need to keep it neat and chronological. That’s because if they are doing poorly, I may want to point out the fact that they haven’t done the homework, or haven’t done it well, that I can use their work as a jumping off point for a conversation, about what’s working for them and what isn’t, in their learning practices.

3. I have problems with the fact that everything is so broken down. Where’s the higher level thinking? Where do students draw connections themselves? Unfortunately, when I thought about these questions, I realized that I rarely introduced or had students work on higher level thinking questions in calculus before anyway. We did bits and pieces here and there, but nothing super consistent. We did do a couple 2-3 day problem solving marathons with formal writeups.

I think each quarter, I will do these problem solving marathons and writeups, and simply break down the skills associated with these problems and writeups, and grade them. Having them broken down into individual skills is fully in line with the SBG plan.

4. Because I’m scared of having too much work to do — and I have so much non-math related things on my plate this year — I have limited the times and the number of skills students can reassess.

5. To get students to think about their own learning processes and styles, what works for them and what doesn’t, I’m having students reflect on why they might have done poorly, and what they did to rectify the situation, before they can reassess. As my Upper School head said, “you’re coaching them in metacognition.”

6. Only the latest skill score counts. I had some debate — highest of the last two scores, average scores, etc.? But I came down on the side that skills have to be retained. And historically except for final exams, I never really taught, demanded, or tested retention. So this is a huge shift.

7. I am going to try to assess most skills twice. But it’s not going to be possible for all.

8. Most importantly, I’m going to go slowly with this, and run with the punches, this year. I’m not worrying about being perfect, or having the perfect skill list, or finding the perfect questions for assessment. Maybe I do have too many skills, and I should chop the list in half? Maybe I am not focusing on integrating problem solving, or making two levels of problems (easy and hard), or intending to change most of my old smartboard lesson plans or what I do in class? I’m going to take it slow, get kids in a routine, get myself in a routine, see how this works out. And once everything is smooth sailing, then I’ll worry about tweaking the system.

[1] The three big “click” moments that got me on board, and then totally shifted my outlook with this:

1. Grades can go down — and retention is part and parcel of this grading scheme.
2. You have to take the most recent grade.
3. I want all my kids to earn As.

Something I realized about myself and SBG

You want to know something I realized about myself and my transition to Standards Based Grading? It’s something I’m kind of embarrassed by, but I’m glad I recognized it so that I can be super conscious about it.

It’s that one of my big fears, something I couldn’t articulate until now I couldn’t quite place my finger on it, is that all my kids are going to do really well. That I’m going to have all 4s and 3.5s for all my students’ final skill grades.

I work at a school full of motivated kids. These kids are largely motivated by grades. [1] I’m  pretty sure there will be tons of reassessment, until students have 4s on everything, or almost everything.

I think something I’ve been proud of is having a pretty solid distribution: a couple As, a bunch of Bs, a few Cs. I don’t know why I’ve taken pride in this — I think having that distribution showed me I was challenging students the right amount, pushing them as a whole. [2] And although I always told my students I’d be happy if they all got As, and I think I would have been because I don’t make my courses easy and I would have been impressed that they all rose to the occasion, I would have also felt like I wasn’t making the course challenging enough for them. Does that make sense? I’ve always used my grading  distribution to let me know if I’m making the course the appropriate level to challenge my kids without having any of them drown.

So when people say Standards Based Grading is a total reorientation in terms of the way you think about the classroom, I realize this is exactly the sort of thing they mean. Because I’m ashamed to admit that one of my secret worries is that all my students get 4s on everything I teach. I don’t want to be that teacher that always gives As. *shiver*

I’m glad that I’ve recognized this secret fear, because it is TOTALLY AND UTTERLY DUMB. SERIOUSLY DUMB. LIKE SO DUMB. If I didn’t recognize this monstrosity in my subconscious now, I would have sabotaged my whole year inadvertently. Yikers!

Hello, earth to Sam, the point of this new grading system is to focus on getting the most amount of kids to know the most amount of material (and also, importantly for me, to teach kids independence and study habits that work for them). And I’ve made a rigorous set of non-fluffy standards. And if my kids can achieve mastery and retention of those standards, I’m going to toot my own horn as loud as I can. I want to capitalize on the motivation of my students to do well.

So now my new mantra is: I want to be the teacher who gives all As, and I’m going to do everything in my power to make sure my students know that I want them all to get As, and do everything in my power to encourage them to use their newfound SBG freedom and independence to get those As. Because my As in this new framework will mean something. Because my course forces mastery and retention. Because my course is rigorous.

Wow, huge realization.

PS. I have a smaller fear that I won’t know how to grade students well, even with my list of skills and my rubric. Not that I won’t be consistent among how I grade students — I am scrupulous about that now. But that I have assessment questions “good enough” that they can reveal to me the various gradations of 4, 3, 2, 1, 0 from the rubric. I think that’s something I’ll pay attention to as the year goes on. It’s something I have to see in action to see if there is some tweaking to do here.

[1] Many care about their grades more than they care about learning. Although both are sort of tied up, they are not synonymous. But regardless, it does lead to kids actually learning stuff.

[2] I’ve also had a serious problem with teachers who always tend to give As to students.