Author: samjshah

PCMI as a learning community

Note: This post was started in Late July, and abandoned until now in Late August

I promised three posts post-PCMI on what I’ve learned — on math talk, on lesson study, and on PCMI as a learning community. Now that it is late August, I don’t know if I can do all three, at least not immediately. So I am going to focus on the learning community formed at PCMI, via the way the program was structured.

I first found out about PCMI on twitter. I am pretty sure it went something like this:

samjshah: anyone have ideas of good summer programs for math teachers?
tweep1: PCMI
tweep2: PCMI
tweep3: PCMI & exeter
samjshah: did exeter, should check out PCMI i guess

I hadn’t heard of it. And no one really could explain to me what it was, and why it was so powerful for them. Many people on in the edublogosphere talked about going there a few summers. To me, that’s advocacy enough, because if you’re a teacher, and you attend a 3 week professional development program in Utah, and then want to do that again, well, that’s saying something to me loud and clear.

Without getting too meta or analytical or anything, I think PCMI showed us how a serious, committed learning community could look like. It mirrored what we would have wanted our school experiences to have been, and had qualities that we wished our schools had. That’s what was so powerful. We participated in this professional development around creating an engaging and effective math community, while actually participating in an engaging and effective math community.

This is how a typical day would look.

I would wake up around 6:30 and take a bath and read. Okay, that sounds like a strange way to start the day, but this was my own private heaven, because I love reading in the bathtub and I don’t get to read much during the school year. Then I would mosey on over to breakfast, where we have a daily buffet — a veritable cornucopia of muffins and eggs and cereal and soy milk and coffee and bagels and other stuffs. Over breakfast, I would generally sit with other math teachers.

You see, at this point, I’ll interrupt and say PCMI actually has a bunch of parallel programs running at the same time. Researchers have a program. Undergrads have a program. Grad students have a program. University teachers have a program. So we high school teachers are just one of a few different strands of PCMI.

Over breakfast, we’d discuss our schools, math problems, what we did the previous night, books we’re reading, TV shows we love, weekend plans, Justin Bieber, whatever! The whole three weeks definitely had a summer camp atmosphere because almost no one knew each other and so there was always lots to discover about everyone. There were about 60 teachers in the program.

Then we’d rush over to our morning problem solving session, which lasted a hair under 2 hours. There would be quick announcements, and then Morning Shorts. These are 5 minute presentations given by participants. Here’s mine. Then onto problems. We’d sit in groups of 5 or 6 (one of those people was a table leader, but they weren’t privy to any of the problems or solutions we’d be working on beforehand). We’d be given a packet of problems and then just set off to go. Nothing else. No formal lesson. Just compelling problems. Sometimes our group would work independently, sometimes we’d talk, sometimes we’d get off track (but super rarely). Connections were made, informally, independently, at our own pace.

The problems were made by Darryl Yong and Bowen Kerins, and are online here (look for the Hand-Outs section). The philosophy of the class is pretty well summed up by their “rules”:

Bowen and Daryl made each subsequent problem set after watching us, and seeing where we were at. We were going, each, at our own pace. To make sure we could keep up with the course, and not fall too far behind, there was a core idea that was put at the start of each problem set — called IMPORTANT STUFF. We had to get through that (we always did), and then we could just work on whatever. But the next day’s material would only really require us to know the IMPORTANT STUFF.

The problem sets themselves were the most well-crafted set of problems I’ve ever been given, in terms of scaffolding. I don’t think you can see really how these problems are so amazingly scaffolded until you actually work through them. Because you will start seeing cycling back to old material, little hints about connections you’ll be making (no connection was ever explicitly given to us in the problem set — we had to do all that heavy lifting on our own), and a general ramping up to some really frustratingly engaging problems.

They were also really funny. With each of our names included in at least one of the problem sets — which actually gave us a nice feeling to see in print. And lots of jokes and puns.

Examples:

Ha, this was the title of the 8th problem set. Notice the use of our names in the problem set! (And each table had numbers, hence the Table 8 in the title.)

I think I got a stomach cramp laughing at the marginalia. Um, it’s okay if your answer has some p in it? Get it? Also, notice Caro(l)’s name in the problem set? (I cut off the l, but it was there, I’m sure.)

And of course the stupid  math humor I’ve come to love so much.

I would just like to point out that all these examples came from the same problem set! So imagine this, every day! Fun! Times!

Okay, so we’d work, and then in the last 15 minutes, or maybe in the middle, Bowen would talk about what people were seeing, he’d maybe throw up a Sketchpad applet he made, or a photograph of some of our work and have us explain it to the rest of the group. And then we’d move on, the next day.

There was one other thing that made this setup so well, minus the self-directed pacing, the well-crafted problem sets, the ability to collaborate. It was that we would only be with our group for 3 days. Yup, that’s it, 3 days. Then we’d be assigned a new group. The end result? After 3 weeks (15 days), we’d have been in groups with almost everyone attending.

Frequent group switching was one of the ways that I think our community was built so quickly, and so powerfully, in 3 weeks.

After some coffee and cookies (that I refrained from eating, thank you very much), we’d then go into our “Reflection on Practice: Connections to Research” groups. For these, we were put in groups of 5 or 6 again, but then we’d have  a few groups in a room instead of all of them. So instead of having all 60 of us in one room, we’d only have 15 or 20 so. We had two leaders, who led us through various exercises, reflections, video-watching and transcript-reading sessions, etc. I would write more about this part of the program, but we learned a lot about a lot of things, so it would be like trying to capture a hundred butterflies fluttering about in a room. Possible, but time-consuming.

For me, the most major theme we hit upon was “math talk” — the purposes of it, how to encourage it, how to evaluate it, and the rewards of it. That’s too big to tackle for me in this post. I’ll leave it at that, until some future time.

We switched up groups and leaders in “Reflection on Practice” session every 5 days.

Then lunch. Mostly we were allowed to sit where we wanted, while we enjoyed more (really good, really filling) meals. At least for me, most conversations were based around teaching, since we had just been given lots to think about in the morning. A few times, we were assigned lunch tables. At first the thought irked me. I’m an adult! But the point was to mix all the different programs together, so we could talk to undergrads and university teachers and everyone inbetween. It was actually fun. For the most part the conversations were enjoyable and engaging, and the few times they weren’t, they were benign and innocuous.

Finally after lunch, we’d embark on the third major part of the day: our individual project groups. There were six different groups (and within some of those groups, sub-groups!):

Reasoning from Data and Chance
Exploring Discrete Math
Investigating Geometry
Implementing Lesson Study
Visualizing Functions
Learning the Math of Image Processing

My group had 9 people in it (2 of them group leaders), and focused on Japanese Lesson Study. It was here we got to create. All groups created some final product. And dang if it isn’t fun to create with other people.

My group went through a sped up cycle of Japanese Lesson Study. We put all our work on a wiki (slightly messy, since it was used a lot), focused on achieving these three goals:

  1. Students will develop a conceptual understanding of converse, inverse, and contrapositive statements, and will be able to use multiple models (specifically Euler diagrams and sentences) to reason about these statements.
  2. Given an assumed true conditional statement, students can distinguish and clearly explain the truth values of the inverse, converse and contrapositive statements — using counterexamples to show the falsity of statements.
  3. Students will develop an appreciation for the precision of language, and usefulness of if-then statements.

We actually got to teach this lesson twice (once to other teachers, and once to real students), and revise it once. One of the most memorable and exhilarating moments at PCMI was watching the lesson that we argued and slaved and nit-picked over come to life when taught to students. The kids were engaged, and I could see them slowly come to understanding on their own. We had come up a list of possible confusions and a list of strategies to employ if they happened. Watching that unfold was… well, you could see the impressive class that results from using collaboration, backwards design, and a focus on student understanding. And seeing kids smile, and work through frustration (productive frustration!), and get that deep a ha moment, that was powerful.

It wasn’t a perfect lesson, but it was perfect enough. A thousand times better than what I can produce in my own classroom. It was an example of the type of lessons and the type of teaching I want to do, where there is less lecturing, serious math discourse, and the  teacher is merely a guide while the students are in the drivers’ seats.

I gave a little spiel about the use of Wikis in lesson study at the end of PCMI. Here is a clip:

Much like our teaching in the classroom, the nature lesson study is organic and evolving. Fundamental to the lesson study process are the dual ideas of: collaboration and continual improvement. The wikispace provides a well-suited home for this sort of work. When we meet, we never quite know what ideas will jump out, what we are going to pursue, what ideas will become central to the lesson, and what ideas will be jettisoned. But it’s important in this type of collaboration – where the creativity of multiple minds comes into play – to have a way to organize these ideas.

As I said, each working group created things which we shared with each other. And seriously, it was all amazing stuff. And then, at 3 something, we were done. That was the end of the required part of the day.

In the evenings, there were a number of fascinating lectures, informal discussion groups, formal classes — all optional, most interesting. Evenings were the social times, where we would BBQ and go into town and watch Veronica Mars and karaoke and eat at a restaurant and play RockBand and go on walks/hikes and …

And the days would repeat.

I know we had a self-selecting group, but it was a learning community at it’s finest. The program was heavily structured — as you can see, it’s a full day. And you couldn’t skip classes or arrive late. You didn’t get to choose groups (except foryour working group, kinda). But within those strict parameters, you had an informal, playful, intense, passionate atmosphere. To me the most defining features of PCMI was the group work — which had groups switch constantly (some every 3 days, some every 5 days, but our working groups stayed constant). We also had lots of different activities (we weren’t in a lecture hall all day) which broke up the time. We were given breaks and informal times to talk about what we had learned. We weren’t really lectured to at any point; it was about thinking and sharing, reflecting, and collaborating. We learned by doing, not by being talked to by Almighty Gods of Teaching. Most importantly, the program was designed for us to be engaged. Clearly. The designers made a conscious effort to be rigorous and interesting. It was differentiated, good for people at all different stages in their careers. And it worked.

If it tells you anything, I spent 3 of my 10 week summer vacation at PCMI. I am going to forego applying to the other program that I have been coveting, in hopes that I can return to Utah next summer.

I’m jumping into the SBG waters! Hope there aren’t any sharks!

Guess what, ma? It’s taken me half a year of mulling, some cajoling from the “inspiring ideological cult”, and the realization that even though I think I’m teaching responsibility, I could be doing way better. So here I am, naked, standing before you… wait, no, that’s not right at all. I have clothes on. Scout’s honor.

Here I am, standing before you, newly self-inducted member of the Standards Based Grading (SBG) cult.

I can’t roll it out for Algebra II next year, but I am plunging — head first — into standard based grading in Calculus.

I made a list of skills that I taught last year — maybe it’ll be of some use to someone out there:

This ordering and skill set probably won’t be changing much for the upcoming year. But it will definitely have to be rewritten for the SBG skill/topic list.

I wasn’t going to blog about my SBG system until it was done, but someone (forgive me, for my mind is weak, and I have forgotten who) mentioned that it might be useful to watch the process unfold. Plus, I have a bunch of questions.

Here’s what I’ve definitely figured out:

1. I am going to assess most skills/topics twice.

2. The skills/topics I won’t assess twice are mainly “explain this idea, statement, or claim (using words, diagrams, tables, graphs)” questions. (Students can reassess those questions on their own, if they want.)

3. Students will have to email me by Sunday night to be able to reassess during study hall on Tuesday, and students will have to email me by Wednesday night to be able to reassess during study hall on Friday. This way I have time to prepare for these individualized reassessments, and students won’t have to individually work on tracking me down.

4. I am not going to be including homework in their final grade.

5. Students keep a binder with all their assessments in it — so students can have them to study from, and I can ask them to see them if I need to.

Here are where I still have to make decisions:

1. Do I want my gradebook to have skills listed, or topics listed? This is a big one! David Cox says this is a false dichotomy, and I can buy that — because skills and topics are really part of the same tangled net. Or two sides of the same coin. Or some other cliched metaphor. But I guess I still think in these different terms. A list of skills, and a list of topics, seems very different to me. Skills tend to be more specific, while topics tend to be more “umbrella”-y. I am leaning towards skills, because that’s where I’m comfortable.

2. Do I want a bunch of short assessments given frequently, or regular (longer) assessments? I think I’m going to go with the shorter assessments, even though it is going to be harder for me to do because I usually have a plethora of students (read: more than 50%) with extended time. I have to figure out a way to not spend too much class time on these assessments.

3. When I give assessments, I might have a few problems testing various cases of something. For example, I might put four problems asking for the limit of rational functions at infinity. Or eight derivative problems asking to apply various skills (e.g. product, quotient, sum, difference). How do I combine these multiple problems into one score? I’m leaning towards a holistic approach, using the rubric, and a lot of feedback.

4. Do I require students to demonstrate/explain to me what they have done to fix gaps in their understanding, in order to be able to reassess? Would setting up the expectation that they need to have done something before they reassess, and then having a place on the reassessment for them to write what they’ve done to fix gaps in their knowledge, be enough?

5. A student’s grade on a topic/skill will either be the average of the last two scores they earned, or the average of the top two of the last three attempts. I’m leaning towards average of the last two scores they earned.

6. Do I allow myself to throw “old” skills on assessments? Like, if students are taking an assessment on derivatives, and I throw on a limit question, is that kosher? This rubs me the wrong way. When I did this in Algebra II in previous years, I told my kids I when I would be including older skills, and I would give them a general idea of what the problem would be on (e.g. absolute value equations and inequalities).  Does that seem like a fair compromise, or is that spoon-feeding too much? I am leaning towards including older material, but with a general warning. It just rubs me as being fair and clear. And I do want students to know that retention is important.

7. Should some skills/topics be worth more than others? I’m thinking of making almost all skills/topics worth 5 points, but I think I might highlight a few and make them worth 10 points. Specifically, I’m considering something like: “Apply the sum, difference, product, and quotient rules for derivatives.” Alternatively, I can break it into two 5 point skills, making one “Apply the sum, difference, product, and quotient rules for derivatives of basic functions” and “Apply the sum, difference, product, and quotient rules for derivatives of more complex functions.”

8. Even though I am not including homework in a grade, I do want students to keep their homework organized someplace, so we can refer to it together. I want it to be powerful — when a student doesn’t do well on a skill, and then we can look it up. If they haven’t done the problems, it will be clear what they need to do to improve. If they have, we can use that as a starting point for a discussion of why they didn’t do so well. So how do I get them to keep their homework, and keep it organized?

So there is where I am. Providing any and all advice and thoughts in the comments would be SUPER welcome!

Always,
Sam

Lost Faith

D.I.G. asked in the comments a few posts ago:

So, Sam, did anyone say what happens when you lose that faith?

People still tell me that they think I’m a good teacher, although I think it’s less and less true as time goes on. I no longer know why I do this job. I haven’t given up yet, as witnessed by the fact that I still have things like your blog in my feed reader.

I’ve been at this career for 20+ years. On some level, I still think I’m probably better in my position than some random person who might be hired to take my place if I left — I have no doubt that I’m basically competent, and not everyone is — but that doesn’t make it much easier to keep going. Did anyone address how to get back to feeling like what you do matters when you’ve lost the faith?

We hadn’t talked about this, and considering where I am in my career (read: early), I had no worthy advice to bestow. so I emailed Peg Cagle, the person at PCMI who talked about faith. She’s been teaching a while. Although she has been busy all summer, she took the time to jot down a few “non-linear musings.”

1.) Faith of any sort demands courage. Courage to believe in something for which there is little if any substantiating evidence let alone proof. Unfortunately, I have no particular insight into the creation of personal courage.
2.) Faith of any sort needs to be nourished. People with religious faith feed it by spending time with other like-minded people discussing and studying their beliefs. The same is true for faith in the work of teaching. Beyond the tools that I gain from attending conferences, talking with colleagues or reading independently, I also renew my belief that teaching is a worthy intellectual endeavor and that by engaging in the work of teaching, I make a difference.
3.) Faith of any sort can be strengthened through challenge. Don’t be afraid to profess your beliefs about public education. While you may not be supporting a popular viewpoint, standing up to a modicum of contrary perspective can re-affirm your own values.
4.) Faith allows for forgiveness. Everyone has weaknesses and doubts during a lifetime of beliefs and don’t beat up on yourself for sometimes thinking that you are delusional for believing that teaching makes a difference. At the same time that you need to remember that our work is an investment in the future, don’t expect to see the long term pay-offs. Focus on the small victories; they exist. And they can get you through another semester, another day or perhaps just another class period.

Thank you Peg, and I hope you find your lost faith, D.I.G.

Personally, I’m not at the place yet where I have started to develop that deep faith in what I do, but I’m sure it will be powerful when (if) it happens.

Not all of us have Soft Skills

This is my post for Riley Lark’s Virtual Conference on Soft Skills

Three. That’s how many different times I’ve started writing this blasted thing. I actually finished a two thousand word post that I almost published. It didn’t really go anywhere – but meandered with fake conversations and some overarching principles and then fizzled out right before you would expect an explosive, pee-releasing BANG.

Because – and you may find this hard to believe – I have nothing to say. Nada.

Even writing this introduction is a way to avoid saying what I have to say. Zilch.

So I’m starting over. On the date this post is due. Here’s the problem.

I have no grand philosophy. I don’t garner the respect of all my students. Some like me, some pretend to like me, some dislike me. I am not beloved. I am not universally hated. I am awkward. I am not a master of soft skills.

When I am in the hallways and I see a kid, I wave and smile and say hello. I actually do this obsessively, and from a distance. Sometimes I secretly think I resemble Miss America. On a float, hand raised high, fanning left and right, as I rumble by the crowds. Just far enough away to feel safe, just smiley enough to say “I care! From! Over! Here!” When I walk with a student, or sit down to have breakfast with a student in the Commons, or have smalltalk in homeroom, I tend to be… a little… well…

Mr. Shah: So, how’s math class going this year?
Stu: Pretty good.
Mr. Shah: I like to hear that.

(awkward silence)

(continues)

(yes, one more second here)

Mr. Shah: Watch anything interesting on TV lately?

(Sometimes that last line might say “Read any good books lately?” or “How do you like the new Justin Bieber single?”)

So first off, I’d like to say to all of you out there reading the virtual conference on Soft Skills, feeling like (a) you are sucky, (b) you don’t make every kid feel like they are one-of-a-kind-and-special, and (c) doo doo…

You probably are…

…and I’m right there with you.

At the same time, I don’t think I have to be awkward forever. I bet these “soft skills” are learnable and I’ll get there.  I’ll get to the point where kids dump Gatorade on me at the end of the year [1].

To avoid having virtual tomatoes – or real tomatoes for that matter – pelted at me, I scrounged up one concrete thing I have done in the past that might fall under the “soft skills” rubric.

It was maybe my fifth or sixth day of teaching. Ever. The Smartboard was broken and I had to improvise my Algebra II class – holy crap holy crap holy crap holy crap. Trust me, this is not one of those “and then I learned I had it in me all along” stories. I SUCKED. Not a “I wish I had said this instead of that” sucked, but a “I left my kids confused, drooling, grunting meeee do not get. brain hurt. hit me over the head with that textbook and put me out of my misery” sucked.

That night, I called my sister upset. She said “take a mulligan.” After I asked her what the heck that was (FYI: me:sports::oil:vinegar), I replied, “better yet, I’ll ask them permission to take a mulligan.”

The next class, I got on my knees – and pleaded for a second chance.  I delivered a (practiced) passionate, funny, histrionic apologia . I followed it with a killer lesson.

The following year, I was teaching quadratics, and I was running out of time. (Aren’t we always?) At the last moment I was asked to teach applications of quadratics — some crazy word problems. I came up with a plan to have students work in groups and present solutions, and it was going to be short and sweet, and then they would get an open-note take home assessment. It was a plan. It wasn’t a well-thought-out plan. Which consequently means it wasn’t a well-executed plan. On the take home assessment, my students got Cs and Ds and Fs. Like all of my students. I don’t mind when I get a bimodal distribution. But this clearly wasn’t their fault. It was my fault. I did a terrible job.

Enter mulligan. I started the next class apologizing. I told them I had screwed up royally. I told them that I thought I had a good way to teach these quadratic word problems, and it clearly wasn’t. And so I failed at my job, and in this case, I failed them. I had a conversation about what went wrong in the way I planned the lessons. I also cancelled the assessment grades.

Honestly, it didn’t feel sheepish going in front of them, admitting weakness and failure. (I thought it would.) It felt good. Real good. And my kids appreciated and accepted my honesty.

Awww, Mr. Shah is a person.

Awww, Mr. Shah makes mistakes.

Awww, Mr. Shah makes up for his mistakes instead of blaming me.

Awww, Mr. Shah is trying to do right by us.

Awww, Mr. Shah respects us.

Awww, Mr. Shah is on my side.

Now, like in golf, you can’t pull this one out of the bag of tricks all the time. But there are moments when it is can really help. And not only your relationship with your kids, but also by giving you a breather – a second chance to right some wrong.

It goes hand in hand with something I put on my course expectations.

I feel like I have a contract with these kids. I expect a lot from them, so I want them to expect a lot from me. They are accountable to me, so I am accountable to them. Not to my school, not to my department, but to them. I also never want them to feel alone – backed into a corner where they have no one to turn to. I want them to know they can always turn to me, because we’re in this boat together.

I got this philosophy from my beloved English teacher in high school. We’d get our essays back with three different colored checks on the front page, and copious amounts of feedback. One day, well into the school year, I asked him what those colored checks meant. He replied, “I read each paper three times. The first time to get a sense of the argument. The second time to give you feedback. The third time to make sure I was fair grading and wasn’t affected by a bad mood. The checks help me keep things in order.” He then followed with the thing that stuck: “I know you guys put a lot of effort and time into these. I want to make sure I do the same.” By respecting our essays, he respected us.

Now, here’s where things get hairy. I write this, and it has that “awww, shucks” ring to it. I remind my kids that I’m always on their side. I show my kids I am human and can connect with them. But remember the beginning of this post, where I started. I don’t have soft skills. Miss America on a float here, remember. I have failed.

Reminding kids that I am always on your side in words and (more importantly) in actions doesn’t always work. Sometimes I can’t reach my kids through an encouraging email, or working with them individually.

Very occasionally a kid comes with (or develops) a chip on his or her shoulder, and when presented with my teaching style – all founded on being clear, consistent, and fair – they shut down or they act out. Most of the time, they haven’t dealt with a teacher with firm expectations and firm follow through. (I don’t have a lot of wiggle room in terms of my expectations.) They’re teenagers, and they resist [2] – and I refuse to lower or change my expectations for them. There inevitably comes a battle of wills.

It’s at this point that I realize that as these students dig their heels into the ground, and as I dig my heels into the ground, I have failed. Because the mantra I am always on your side doesn’t apply. We don’t have a common goal anymore. [3]

Consequence? Neither of us win – both of us lose.

I can count these students on one hand, but they can (and have) defined the atmosphere of a class. That sucks, because who wants to have a class which you like going to less than your other classes? I don’t have a bag of tricks for these kids yet. The unreachable kid – I don’t know how to reach ‘em.

This is where I am at in terms of soft skills.

So yeah. I didn’t want to end this on a “feel good” note. I want to say that soft skills are hard, and they don’t come naturally to everyone. The fact that this was so hard for me to write shows me that it taps into some serious insecurities I have as a teacher. But they are important, because they keep you and kids on the same side. You’re a team. When you’re not on the same side, you need to ask how is it that you can be on the same side again. But this isn’t a feel-good-I’m-great post. I can’t always say I’m successful.

So to all of you who are sick of reading how awesome everyone else is – me too. But let’s make a pact. Let’s consciously work on it to make things better.

POST SCRIPT

Other things I do that might qualify as soft skills:

  1. I always keep candy on my desk, and if a student comes by (for almost any reason) I’ll offer ‘em some. I’ll often offer students I don’t teach candy too, when they come in the office and look a little down or worried or anxious.
  2. I write a single letter at the end of the year and distribute it to my seniors (but  I address each of the letters individually and print them out on school stationary and sign them and put them in school envelopes). The letters are sappy, chronicle our year, and give unsolicited advice.
  3. At the beginning of the year, I have students fill out a google form with some basic information about them, their calculator serial number, and with some questions they must answer from the course expectations. I also ask what they are most worried about. I then write, to the students who say things other than “nothing,” individual emails to these students addressing their concerns and reminding them they always can come to me if things get stressful.
  4. On the first few assessments, I’ll have a question where I have students write about what they’re feeling about the class, or about the material, or what they’re nervous about, or what they still don’t get. It’s all pretty open ended. I write short responses to each of these.

 

[1] I don’t remember who said that. But it’s an evocative image, no?

[2] Heck, I’m known for being resistant too. It’s not just teenagers.

[3] To be clear, it’s not like I just leave things be. I meet with these students individually a few times to see what’s going on in a non-confrontational way, I ask the student “what can I do to help things? I want us to be on the same side again.” I make our conversation a give-and-take, without lowering my expectations or giving the student things I am not willing to give every student. I bring the adviser or dean into the conversation. I engage the parents. Usually these things work to some degree. But there are those few kids where nothing I do works.

Powerful Talk by @profteacher

So I want to a session led by @profteacher today at 4:30pm. I gave a big shoutout to him on July 11th. He’s a university professor who used his sabbatical year to teach high school math in a public (urban) high school. If you didn’t check it out then, check it out now.

This was one of those emotional talks for me to listen to. And I’m not an emotional person (unless Oprah is on). It was about being a first year teacher. The defeat and the joys and simple observations. I say “talk” but it actually became a rich and compelling conversation among, I don’t know, 30 or 40 dedicated teachers — all at different levels of teaching. It was raw and honest. It wasn’t defeatest or idealist. It was real.

There were two points that were made, that sound like sound bytes. And usually, I’d just brush them off as general platitudes or something. But I know and trust these people, and in this context, these points were deep and rich and I think I’ll probably treasure them.

1. “Teaching is the connsumate act of faith — faith in what you do.” One participant said this, an experienced teacher who talked about how the emotional part of teaching evolves, and after a number of years, she started really believing in this. She continued to say that you won’t be there when a kid gets a college acceptance. You just won’t know how and with whom you made an impact. (In fact, the student might not even be able to recognize it.) That’s where faith comes in. Faith that what we do matters.

2. The presenter said his one big take away from this first year: you need to have students know and feel that they can be successful. The lessons don’t have to be exciting — they can be routine and boring. “Factoring worksheets!” he said, “they will start tearing through them because they know they can be successful.” His discipline problems disappeared when he discovered this. How to do that? Developing lessons through careful crafting and scaffolding just enough — so that students are going through “productive frustration” — where the next step is just within reach. Again, just words. Words I would ignore, if the presenter hadn’t just developed and delivered a curriculum to me for 3 weeks which embodied everything he said. Scaffolded. Carefully crafted. And there was … everyday … engaged, productive frustration.

I’ll write more about that later. But I just needed to jot these two points down in the spare 10 minutes I had before dinner.

Exasperating Problem

So a while ago, I mentioned to some of you on twitter that I was getting really frustrated with a particular problem we were presented with. I have a conjecture that I’m almost certain is true, but I can’t prove it.

Consider the unit circle x^2+y^2=1. Plot n equally spaced points on the circle starting from (1,0). Now draw the n-1 chords from (1,0) to the others. What is the product of the lengths of all of these chords?

(There is an extension problem, which is changing the unit circle to an ellipse 5x^2+y^2=5, for those who already have seen or find the original problem too easy.)

So feel free to write your own blog post with your solution, or throw your solution in the comments (just write SPOILER at the top so we know…).

What I’m interested in is if we could get a precalculus class to get the solution to this problem. Where they actually understand it. So if you had, say, 15 non-honors precalculus students and one week to work on this problem, how would you design the lesson?

I guess you have to have solved it or have seen a solution to know how to design the lesson. But even if you didn’t solve it (a la me!)… if there’s a solution you’ve read that someone posted in the comments… what would you do?

UPDATE: Mr. Ho has a great GeoGebra applet at his site; Mimi has some nice colorful diagrams and some explanation up at her site. Also, for those who want to wording for the ellipse problem… This extension I haven’t seen before, so I am citing Bowen Kerins (see comments below!) or Darryl Yong: “Take the diagram you drew in [the unit circle problem] and stretch it vertically so that the circle becomes the ellipse 5x^2+y^2=5. All the points for the chords scale too. What is the product of the lengths of all of these chords?”

Looking past teachers to teaching

Today I attended a session where three university profs — ed researchers — formed an informal panel. There was one important point that came up at the beginning, and became a riff for a few minutes. It was, as you prolly suspected from the really innovative title of this post, about the power of looking past teachers to teaching.

It’s a slight distinction, but crucial to the reorientation that I’m having about teaching.

Some points that came up in the conversation:

  • Replacing teachers won’t change things; replacing teaching methods will.
  • Focusing on teaching and not on teachers is the basis of lesson study (and the Seattle video club I talked about in the last post). It focuses the conversation on teaching/teacher moves.
  • Changing the conversation from teachers to teaching more readily implies that teaching is learnable. So we have to look past individual teachers to the methods of teaching. That being a good teacher can be taught. Another way to think about it: teaching is a complicated activity, rather that something owned by a particular person.
  • There are universal tasks to teaching that we can investigate (e.g. which ideas to privilege in a classroom).
  • It gets us away from the “I taught it but they didn’t learn it” phenomenon. That phrase doesn’t really make any sense when focusing on teaching and not the teacher.
  • The greatest untapped resource we can use in the classroom are our students and their insights. And by focusing less on the teacher and more on teaching moves, we can tap into that.
  • This outlook shifts the conversation away from teacher bashing (but one should also be cautious of going in the other direction of student bashing).

Yes, I know. There are some inconsistencies, and worse, this is all very abstract. And I HATE THAT. But this all tapped into the idea I wrote about recently, about how teaching moves are something that one can pay attention to. One can learn. One can revise. And through this process, hone the craft of teaching.

In other words, the focus on teaching instead of teachers is that it puts the emphasis on the ways teachers can do their jobs by focusing on students and learning.

So that was one part of the talk. In another part of the talk, there was a question about the constant tension between the jam-packed curricula with a zillion micro-pico-standards and getting students to really grapple with big ideas.

One speaker said that we “need more effort and courage” from teachers. I drew a sad face in my notebook next to that.

The second speaker actually spoke articulately, in defense of having common standards in theory [1]. He also said that he doesn’t see the problem as having a zillion pico standards. It’s that we go through all these little ideas that never get added up to any big ideas. His suggestion for dealing with this is to outline learning trajectories, with big ideas as the landmarks on the way. I don’t know what precisely he had in mind, but I figured that it probably involves student drawing connections by working on unfamiliar problems that force relationships among mathematical ideas (e.g. systems of equations with matrices; asymptotes for the tangent graphs and asymptotes of rational functions; absolute value equations and absolute value inequalities; etc.).

The third person then finished up speaking about the Common Core Standards — and eloquently continued the second speaker’s defense of standards.

That’s about it for the maths stuff I want to write about. (It’s late and I have lots to do tomorrow.)

On the non-math side of things, I had a wonderful night BBQing with friends and watching the sky change hughes, from orange, to light blue, to dark blue, to black. As the air got colder and the light retreated, the stars starting coming out, first slowly then quickly. As people left, conversations got less frenetic and more personal. And I left, after being regaled with a shooting star, at peace with Utah.

[1] Having these standards gets us focused on teaching. It also promotes the sharing of ideas; if someone gets it/does it right, then those lessons and approaches will be in demand.