Carl's Teaching Blog

A place to talk about teaching and learning

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Clog: Suddenly a math argument breaks out

So I always want to try to get kids to have big mathematical discussions in class, but it doesn’t always happen. Today a VERY lively mathematical discussion broke out in class. There was a point where like 3 kids were up at the board, vigorously gesturing at the models that were written at the board, while the rest of the class waited breathlessly for one of them to be conferred as “the answer”. It was out of hand.

Now this didn’t exactly go well. I’m not saying a bunch of kids screaming at each other is ‘productive discourse’, but that’s to be understood. The class was actually a class I was subbing and it was the second day the class had ever met, and a huge chunk of kids missed class on the first day. Class started very quiet as to be expected from a new group with a sub teacher, so I wasn’t really emphasizing turn-taking and sharing, which I would later regret.

Here’s the problem that the class was working on:

Draw a diagram 1:

There’s a softball league with three teams, The Alligators, The Bears, The Crocodiles, The Dolphins, The Eagles, The Foxes, The Grizzlies. Each team plays each of the others 3 times. How many games are played?

I asked the students to read the problem as a group. It would have been nice to stop after the first sentence and do a Notice, Wonder, but instead I asked a few questions to make sure people were interpreting the question right.

  • “What’s important here?”
  • “7 teams, 3 times”
  • “What’s a game?”
  • “When two teams play each other.”

They started working, and when they did I made sure I only commented on their process, and I didn’t confer if anything was right or not. Anyone who thought they were done I asked them to explain their process further, or to draw the diagram that the teacher requested. At this point the class was pretty low energy, and seemed to be convinced that they had the right answer.

The was a quiet girl in the middle of the U shaped tables whose method I wanted to talk about first. She literally listed out all of the games, and the rest of the kids just immediately started multiplying. The Elmo didn’t work, and the class was pretty low-energy, so I started by writing what I heard her say, which also allowed me to organize the work slightly (re-writing student work isn’t ideal, she should have done it, but it made sense given the context). Once all the games were up there, grumbling started.

  • “What happened after the AG team?”
  • “Why is there no BA game?”
  • “Why doesn’t G play any games?”

After first thinking that she made a mistake, I encouraged the student to defend her work and explain that after we counted the first ‘AB’ game where The Alligators played The Bears, we didn’t need to count The Bears playing the Alligators. Grumbles.

One of the students who disagreed offered to describe what he did. He said that The Alligators are going to play 3 games against The Bears, 3 games against The Crocodiles… and thus they would play 6 games 3 times, or 18 games. So then the rest of the teams would play 18 games. So then a bunch of people agreed with that, but there were now contrary grumbles about the games being double counted. Around this point people stated asking me what the answer was. I said that’s your job to figure it out. If it’s the answer you should be able to defend it. THe students kept talking about it and as they talked I came up with representations to write on the board to show what they were thinking. I drew a table with ABCDEFG along the side and the top and wrote the number 3 in all the spaces so people could see all the games. As people began to question the number of games, I wrote out the decreasing cascade of games 18+15+12+9+6+3+0 as the student in the back who listed the games began a more vigorous defense of her ideas. The main student opposed to this still had questions and offered to come to the board and draw his own diagram. I stepped to the side. Suddenly 3 students were up there having a screaming match and the rest of the class was following along vigorously. One student had defined “playing” as both hosting a game, and travelling to play a game, and so he was counting the double-counted games. I explained how this misunderstanding of the problem led to his different understanding, and if that is the way it was defined, then the problem would have a different answer. The kids were still wanting me to pick an answer, but I think if you understood the problem differently and can explain your work, then that would be your answer. The class quieted down and everyone said their brain hurt.

Why is it so darn hard to push ‘Send’???

Many people say, I’m talkative. People rarely describe me as quiet either online or in real life. Despite my 100+ blog posts and my 3000+ tweets, I feel like frightened every time I hit that blue button that spreads my thoughts all over the internet. I’ll still read some tweets and then think about replying for minutes, hours, maybe days only to decide not to send anything at all.

Why shouldn’t I just hit send already? Well some of the typical reasons that float through my head are “I don’t really understand enough to comment on this” or “I already know so much about this, that they couldn’t really care what I think.” or “I have a pretty good response, but it’s not JUST right. Besides, this person basically said the same thing I said.” These are excuses are pretty thin. Too thin to hide behind, infact. The reality is these excuses are just proxies for the general fear that goes along with any kind of public exposure, be it twitter, a college party, or my 9th grade student council elections.

Today I spent a lot of time thinking about my inability to hit send after reading this tweet from Dan Meyer:

Now I didn’t reply to Dan’s tweet, but not because I wan’t interested in it. There are lots of reasons people question posting something, more than just whether it’s “ok.” Possible replies rolled through my head all day but, naturally, nothing congealed into a 140 character response. In the moments before I made the choice not to hit ‘Send’ here, as per usual. Specifically I thought:

  • Did somebody else already post this somewhere else?
  • Is this conversation still active?
  • Do I actually know enough to add to the conversation?
  • Are my thoughts too big to fit into a tweet?
  • Are my thoughts too small to justify a blog post?
  • Is there something better I should be doing then sitting here vacillating about whether or not to hit ‘Send’?

Is that just me? I do have weird brain chemistry after all, it’s hard to gauge what’s normal. If these do float through your head that’s good, they are valid. They haven’t gone away for me, and the reality is, they’ve been there when I don’t hit send, and also when I do. These thoughts are always going to be floating through one’s head before they put something out, but that d0esn’t mean people should hide behind them. These kind of thoughts should be considered, and used for improvement, but not render you to silence. Let me just go through and respond to these thoughts so you won’t have to next time (You’re welcome).

  • Did somebody else already post this somewhere else? – Well, this isn’t as important of a question as “Did YOU post about this yet?” YOU are the only person in the world working in your context, with your population, and your ideas matter.
  • Is this conversation still active? – If you’re saying something relevant, a new conversation will start. If not, you just got the last word.
  • Do I actually know enough to add to the conversation? – If you don’t, the best experts in the world may read your tweet and point you in the right direction. Besides, if that’s your concern, not posting means you’re just going to be stuck there with your ignorance.
  • Are my thoughts too big to fit into a tweet? Are my thoughts too small to justify a blog post? – Well, write the post and tweet it. Or, write a bunch of tweets and put it on your blog.
  • Is there something better I should be doing than sitting here vacillating about whether or not to hit ‘Send’? – Nope. If it’s important enough for you to think about it for this long, it’s only going to help your brain to continue thinking about it with friends on the internet.

Hopefully this will might help others who hesitate before tweeting and blogging. I was in the middle of hesitating about writing this, but then I started watching this video about a guy who was hesitating about posting a song to soundcloud, but then he did and it ended blowing up. Sure this guy was a talented artist, and he made a really good song. You can hear his doubts in between the rants of Gary Vaynerchuk who owns the youtube channel. Gary V is a social media guru and one of the big things he talks about is the importance of just putting your stuff out there and letting the world decide. Seemed relevant.

Beyond Linear #NCTMAnnual 2017 Presentation Materials

So the time came for my #NCTMAnnual talk in San Antonio. I had been thinking about it for weeks, telling my co-workers and family about it, and furiously touching up my slides. When the time came there was no one in my room. Not even me. I was in a cab travelling uncomfortably fast towards the conference center a full 25 hours after my planned arrival in the city. At that point I was already aware that I wasn’t going to talk, and it was pretty disappointing (and the cab driver going 80 was not helping). The NCTM was hit with a fortuitous wave of cancellations and a slot emerged at the end of the day Wednesday where I could give my talk.

My talk was a 30-minute burst so it began like most bursts, with a quickly paced race through as many ideas as time would allow. This talk is about a way to think about quadratics that I have been thinking about after the work in my school. At a transfer school, students will enroll in your class at any point of the year. Like, right after you had your really great introduction to functions, for example. Or perhaps right before you teach your unit in quadratics. I began to think that I should teach quadratics with the same focus usually reserved for linear equations. We focus on starting with exploring places where quadratics exist naturally, and taking students thoughts about those patterns and connecting them with the graphical, tabular, and equation-al(?) representations of quadratic equations. This is different than the last textbook I used which opens up the quadratic unit with F.O.I.L.

The session was moving along pretty smoothly until I gave the participants the washing dishes problem in the slides below. The problem was meant to be a quick taste of a strategy to offer kids a genuine chance of interacting with a quadratic pattern that arises from a real-world scenario. There was nice hum of thinking mixed with frustration in the room. I moved to cut it short as we were already 25 minutes through my alloted 30 minutes and prepared to post up the “answer” on the slide. When I got everyone’s attention and told them “we’re going to have to move forward” everyone looked at me like I was blocking the television during the fourth quarter. “I really want to hear what you guys are thinking,” I said, “but we are about to run out of time. It’s a burst so it’s only 30 minutes.” Someone in the crowd was said “No, let us keep going…” another said, “Just finish… there’s no talks after this, and we have nowhere to go.” I was floored.

We worked more about on the problem longer while I looked around for an interesting approach to highlight. It became a team effort as we all worked to make most out of our extended time together. Sadie helped me figure out how to focus the document camera and we talked as a group about Janet’s example of work that she ripped out from her notebook. Then I showed a way that student might approach the problem if they followed the linear approach I was describing earlier. Then a full 20 minutes after I was supposed to finish I began talking about another problem that fits this mold, and how these kinds of problems can be created to help kids make sense of quadratics. Janet left saying “I’m going to think about examples that I could show my kids.”

This 50+ minutes of my 30 minute talk has been the highlight of a conference full of highlights. I was honored and excited to have a great group of people to do math with. I’ll have to write more at a later date about the actual math in my talk, but I wanted to write about my experience giving the talk. Thanks to all of the people who came, our 50 or so minutes together totally made up for the 25 or so hours I had to spend behind TSA security the day before my talk was scheduled.

 

Handout:

Download (PDF, 323KB)

Some quick thoughts about Finance and Math

I haven’t written a blog post in a while, so I figured this tweet is as good a prompt as any to get back in to it.

I taught a math econ unit for a few years now and I have a few things that I do. I did a Global Math Dept talk once, but it was the only one ever that the audio recording dropped out on (sad face). Here are the slides from that.

In this talk I pretty much list all of the tricks I have, some of which include:

  • Giving kids credit card applications is always a good time. I’ve collected a box of those mailers that they send me, but you could find some of them online. Then you can make a task where the kids use exponential growth to see if they can manage life with a credit card. Here’s what I’ve used in the past.
  • Stock market games are always fun, even though there isn’t too much math involved unless they are doing some next level modelling and regression and what not. I had kids go to Marketwatch and play a game with the rest of the class to see who can make the most of their $100,000 investment.
  • My co-worker has a bunch of projects he uses for a lot of the formulas that use exponential growth for financial purposes.
  • Speaking of Exponential Growth here is a little lesson on exponential vs. linear growth that is a play on that whole “grain-of-rice-on-the-chessboard” thing:

Is this helpful? Let me know your thoughts in the comments.

How this political atmosphere is changing me (and this blog, and my twitter)

This Sunday feels like a good time to write about what’s happening on my blog. I haven’t written as much as I would have liked over the past few weeks, so I feel a little explanation is due.

We have not had any actual classes for the past few weeks so there has been any actual teaching since my last class log. Most of those class days were spent floating around the room helping kids, so there wasn’t much to blog about earlier in January. There was also a lot of administrative tasks needed for the individual scheduling for students. (It’s a super cool network of google spreadsheet that I may blog about later on my other blog). … Oh and it was my wife’s birthday. So we had our first baby-sat date night at Olmsted, which was really nice. I plan to write more with the start of February, where I won’t be inhibited by my lack of classes to talk about, my administrative demands, or my frenzied last second search for birthday presents. So I guess that’s the only thing that’s going on with this blog about teaching and learning.

Oh wait. I actually didn’t mention that other thing that happened towards the tail end of January. Technically it was a November thing, but most of the work that I took on didn’t start until mid-January. I really underestimated just how much time I would need to devote to this. What I had to do for this thing was curl up into the fetal position on my couch/floor/classroom, scroll through the as much internet news as I could, and try to convince myself that the world wasn’t imploding. This has been taking up pretty much most of my time for the past two weeks. It started just around the time that my colleagues and I watched the inauguration during lunch.

My colleagues and some students watching the inauguration during lunch

Since the inauguration my attempts to figure out what is going on have been voracious but not necessarily successful. I’m diving into all the political news I can handle, both to try to harvest any optimism from news of the resistance, and to try to make sense of a Trump voters view point. Finding actual information is really hard. There are active disinformation campaigns, information that isn’t made public, and my need to parse through boring information so I can know about typical government procedures. This frustration would make me give up. In the past I’d ignore political news and go about life assuming all is well. I can’t do that now. Partially because of the switch that flipped on election and the healthy paranoia I’ve developed about the world around me around me (I think I learned this from my dad, who listened to Rush Limbaugh on the ride home after work every day even though he hated Rush Limbaugh). The other reason I can’t bury my head in the sand is genuine interest. I really do want to know what all of us Americans thinks about America. Why is it so hard to figure out what prior actually think?

https://twitter.com/JustinAion/status/822848618881306624

The day of the Womem’s march I was slated to teach saturday school. The trains were crazy, so I took a cab with a particular chatty cabby. I asked him whether the march was causing traffic problems, and he replied with “Women! Why are they marching??? They already have everything!!!” To which I replied with a swift punch in the face a heavy-handed discussion of places where male privilege operates. When I got out of the cab I was a little disgusted. I was also struck by how little conversations like this happen. This was the only conversation I had about any political issue with someone I didn’t agree with in maybe 8 months, and it was an election year! Even though I’m off of Facebook, apparently I’m still in the echo chamber that prevents me from understanding what’s driving the people outside the chamber.

Over the past few weeks it’s become clear that I am going to need to develop more faith in America if I am going to be my normal, optimistic, not-paranoid self. I also have learned that the media isn’t helpful, and if it is, it will be very hard to find. But at the same time, I’m surrounded by Americans who perhaps help me restore that faith, or at least help me understand exactly how much further we still need to go. Maybe the next step for my own sanity is to start talking about politics. (*gasp*)

Which brings me to my blog, and my twitter. Both of these are venues to have conversations about America, Democracy, and the political actions that shape them. While I am an educator, and am only interested in this blog and the twitter for developing myself as an educator, I can’t ignore isolate my profession from the world we work in. Education is indistinguishably intertwined with the needs of the public and the way that public is governed. We as a people must democratically decide how to fund our schools, where to build them, and what to teach in them. When students arrive, we must show them how this democracy works, give them experience operating within a democratic system, so they can grow up to lead our democracy into the future. This is a lot of work, more than can be handled in a civics class, work that math teachers can and should be sneaking into their classrooms. This could be modeling what it means to be an active citizen. It certainly means having active discourse about important ideas, and finding ways justify those ideas and defend arguments. It means pushing students to see things from multiple perspectives, cooperating on meaningful tasks, and communicating that work with others. This is all just good math teaching. With only a slight lateral shift, we can help students apply these principles while discussing current events, systems of iniquity, or even the ideals of democracy and in the process make our mathematically proficient students into mathematically proficient citizens.

As educators it seems that we should be sharing strategies to live up to these ideals. As grateful as I am to lean on my extended network for real world contexts for multiplying binomials, I am now also grateful in hearing how to talk about the inauguration, or talk about immigration, or whatever the next executive order will be. Learning what they think helps me to gain more perspective, and will increase the usefulness of a class conversation if students want to have a discussion about these things. Posts like Michael Fenton’s recent one are encouraging. As I try to figure out how to inject more democracy into my teaching, my work with teachers, and my work as an administrator, it will be invaluable to have honest perspectives around these important issues as a part of my time line. Thinking about these things regularly helps inform my thinking, and prepares me to facilitate a conversation about an issue if it comes up in class, or in a parent meeting, or one-on-one with a student.

https://twitter.com/MyMathscape/status/824762528110608386
On Thursday I had such a meeting. A student who disappeared for months returned to school to enroll for this semester. This student is not a citizen. I remember a few years ago he was worried about not being able to go to college. Who knows what his worries might be. Based on what I have been reading, however, I knew that New York City was a sanctuary city, and our mayor was prepared to fight against deportation. I told him that there are a number of resources I could share, resources which I have seen shared across my personal network. The student relaxed, signed up for a full program, and is on track to graduate by June.

Being aware of the political concerns of others, and coming up with ideas for ways to help students understand political issues is now ready important to me. I am really glad to have people on my feed that post more than just math. In fact, I hope people could organize their talks about the intersections of math, politics and democracy into some kind of hashtag or regular chat. That’s a pipe dream, right now I’m going to keep trying to figure out what’s going on, and being really grateful to all the people sharing what they learn on the web. 

If that, or any of this sounds interesting to you, PLEASE let me know in the comments.

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