Teaching Procedures.... Is it all bad?

Sometimes it feels like I’m just teaching students patterns that go in students’ short-term memory. In general, students pick up on the solution pattern if you show them enough examples. I think they’re really getting it. They say, “This is easy.” I’m glad because at least they’re not complaining about how hard math is and how they suck at it. Then I come in the next day and they can’t remember what we’ve done. I spend time showing them the pattern again and little light bulbs go on. Then I teach the new lesson, which is usually another type of problem with a slightly different solution pattern. They practice it and get decent at it. I assign them homework that is just like what we did in class. Some of them do it and some of them don’t. They come in the next day and many have forgotten how to do the problems. So I review again. Review is good, but I wonder if I’m just teaching them patterns that go in their short-term memory.

I want them to be getting the big idea of the unit and be able to fit what they’re learning into this grand scheme. But when I was in high school, I didn’t know how things fit together. I learned math by recognizing patterns and mimicking problem solutions for the most part. Sometimes I got a glimpse of how some topics were related, but rarely saw the overall picture. It wasn’t until college when I began learning even more advanced topics that I could look back and see how concepts in algebra were related, for example. Many of the different patterns and procedures I had learned were actually different perspectives of the same concept. (Which is something I think is cool about mathematics.)

I don’t spend time teaching the connections because I don’t think they’ll get it. I also don’t think I have enough time to teach them. I don’t know if these are false preconceptions or if they’re true. I also don’t know if it’s good or bad that I don’t teach the connections very often. I used to think that you’re a bad teacher if you focus on procedures. I still do, in a way. However, today, as I was having this thought about whether I should be more focused on deep understanding and conceptual understanding or on procedures, this blog post by Christopher Danielson appeared on my NetVibes account: “Sometimes in mathematics, we need to live with new notation before picking its meaning apart too carefully…. Patterns are powerful tools in mathematics. Tabitha’s [my daughter’s] experience in the teens gave her powerful intuitions for the twenties” (Christopher Danielson, http://christopherdanielson.wordpress.com). In the comments, someone writes: “cf: John von Neumann: ‘In mathematics you don’t understand things. You just get used to them.’ One of my favorite thoughts about mathematics, and persistence, and playfulness.”

The perspective that learning patterns is fundamental to learning math countered my thought about understanding the grand scheme. I do believe that teaching connections is important in math. But I can’t overlook the importance of the procedures either. Sometimes you learn the procedures or notation first, and once you get good at it, you can then start to see how it relates to other ideas. So thanks, Christopher, for that timely thought!

How Many Squares are There?

I posed the following problem to my 7th hour Algebra 2 class yesterday as a warm up:

How many squares are there?

Most students were interested in the problem and stared at the board for a while trying to figure it out. I gave them some time to work on it and then asked for answers. Many said 10 or 11. I asked for a student to come up and show the squares he counted. He got them all but one, and the class pointed it out. (Answer is at the end of the post.)

This problem was still on the board when I came into Pre-Calc today. The students were instantly fascinated. I didn't even direct them to the problem. A few students at their desks were considering it and as other students trickled in they noticed and stopped to look at the problem before sitting down. By the time the bell rang, there was a semi-circle of students all looking at the board and working on the problem. I wish I could have gotten a picture! If only all math classes were this engaging for students!!

I was surprised with how intensely students were engaged in this problem. I have seen other cases where students are fascinated with puzzles, especially students who aren't very into math class. (In Algebra 1, I supply puzzles for when students are finished with work, but students have started working on them during class, distracting them from notes.) The challenge that I see in this experience is: How do I make math class more like a puzzle? I believe that most people are naturally engaged by puzzles or by challenges that are just within their reach. How do I harness this interest and use it to teach math?

(Answer to puzzle: 11 squares)

An Attempt at Differentiation

Last week in 4th hour Algebra 2 I had the students practicing problems on whiteboards. I would call out and write down a problem and they would solve it on their individual board and then hold it up for me to check. It works pretty well as a means of practice. However, I noticed some students weren't really trying and were mostly talking so I got on their case about it. One of them said, "Well, you're always helping people over there." She meant I'm always helping people on the other side of the room. She's mostly right. I think it takes too long to help her and her friend and sometimes I think they don't put in any effort, so I tend to help those on the other side of the room more, and there are more students on the other side. What she said kind of rung in my ear for the rest of the day.

Then I read this blog post by John Scammell about how he does homework. He teaches a lesson, then gives an exit slip with a couple of problems like they learned that day. He collects them and sorts them by how many problems students got right. The next day he splits students up into three groups - students who got it, students who partially got it, and students who didn't get it at all. He assigns extension problems to students who got it, has students who partially got it correct their exit slips and then work on more problems, and he works with students who didn't get it at all.

On Friday last week many students were mentally checked out of class because prom was that night. We had taken notes the previous day on simplifying and expanding logarithmic expressions, and on Friday we practiced them on whiteboards. I knew that some students knew it pretty well, while others had no clue what was going on. I thought I'd try John's method in class today. It felt risky to me because I didn't know how the students would react to being split up like that. They also don't like to have to move seats. So I gave out a slip of paper and had them do three problems without any introduction. They scored them themselves and then passed them forward. I quickly grouped them and had all students who got 2-3 right work on one side of the room on their homework, which I knew few students had finished. Then I worked with the students who had gotten 0-1 problems right. I had to ask one student a couple of times to join us. Finally everyone was together and we were working on the homework. I answered some questions, and then they started asking each other or just volunteering to help each other. We worked for about 15 minutes before the bell rang, and I felt like it was more productive than a class I've taught in a while. I knew exactly where these students were at and they were getting the remediation they needed. They were mostly focused and working. One student even mentioned that it was helpful. I don't know how the rest of the class was doing, though, which is a slight problem. But overall, I was surprised at how well it worked. This is not a strategy I am going to use every day and I'm not sure that it would work every day in my class, but I am happy with the results from today and glad I risked it.

Natural Consequences

On Wednesday we finished class about 5 minutes early in 4th hour. This class always stands by the door waiting for the bell to ring (which I will never allow when I have my own classroom). They crowd up really close to the open door, almost out into the hallway. They asked repeatedly if they could go to lunch early and I said no. They tell me that the teacher across the hall always lets his students our early, and my students also tell me they always get to lunch last and have to stand at the end of the line. Boo hoo! While they are waiting at the door, I have my back to them working with another student. About a minute or two later I notice it’s quieter than usual. I look over to the door and notice some students have left. I go over and look out into the hall. I don’t see them. All the other students who are by the door look at me with a slightly guilty but I’m-not-saying-anything look. They want to know what I am going to do. I don’t really know what to do. It bothers me that they left early and they disobeyed me, but it’s not a major infraction. So I ask the students what they think I should do. They say I should let them out early the next day and make the three boys who left today stay until the bell rings. What an appropriate consequence! So I agree to it. What’s even better is that the students themselves will enforce it, and they think it’s fair so they won’t put up with any bullshit from the three boys if the boys complain about it. I don’t mind letting everyone go 5 minutes early the next day for this, and honestly they are usually mentally checked out of class the last 5 minutes.

So the next day I let everyone out early except the three boys. I was thinking about what I would do with them… should I make them work or what? I decided I would just let them do whatever they want, and I would clean up like normal after class. They sat there on the desks by the door and kind of talked. One other girl was still in class waiting for her boyfriend, and they mentioned to her that it felt awkward being there by themselves with just a teacher. They tried to tell me it was unfair to let everyone else leave five minutes early because they didn’t leave five minutes early the day before and they were just going to their locker, neither of which are totally true. I just let them continue waiting there while I shuffled papers. Finally, I let them go right before the bell was about to ring.

I felt really good about this consequence. I had class buy-in and the consequences fit the offense perfectly.

You're a Good Teacher!

Yesterday I walked into my first hour class, Pre-Algebra a little late (whoops!). At least I'm not really student teaching in that class so I wasn't late to teaching, just to observing. Still I've been getting lax in being on time, which is not good. Anyways, so I walk in and as I'm coming in the door, a group of students look at me and point and say, "There she is!" I'm thinking, this is strange... What's up? This class is normally very passive in the mornings, and while I've taught them before, I don't feel a very strong connection to this class. One of the girls in the groups says, "We want you to teach today!" I ask "Why?" Mr. Collins says they want to trade me for him. The same girl says, "Because you're a good teacher!" What? Where did that come from? I didn't know they really even paid any attention to me. It made me feel good to think that they think I'm a good teacher. They probably just like a change from the norm, but still it was nice to hear. I also think that middle schoolers see through teachers pretty well, so I take it as quite a compliment that they think I'm a good teacher. This is one of those things you store away to recall when there's no feedback or you feel like things aren't going well at all.

Update on My New Approach to Homework

After the last test on radicals, I decided I needed to change some things. You can my prior post on that subject. Homework is now due every day and points are docked if it's late. More students than ever are doing their homework. We either start off the class with a warm-up review problem or we go over homework. Both are good for review. I am always surprised when I do the warm-ups how little they retain from the previous day!! I now know that they probably didn't learn many topics along the way last chapter and I didn't realize it because I never assessed their learning on the previous day's material. Students now come in with questions on homework and we can address things they don't understand more immediately. I love this system! It's such a basic and traditional method, but it's new to me! It feels good knowing where students are at, hearing their questions, helping them recall information, and being able to address issues upfront, including talking to students who haven't done their homework about what's going on. I'm hoping that this change will improve their retention and understanding and that at the end of the chapter I won't be as frustrated with the results on their tests.

Happy Day

Teaching is a strange thing. The last two days I have been really exhausted and worried that I’m burning myself out. I was in the car for 6 hours yesterday on top of teaching. I’m behind in preparing lessons and wasn’t totally ready for my Algebra II classes today. I have to make a quiz for tomorrow, write lessons, create Algebra I centers, etc. And yet right now at the end of the day I feel really good. I have a lot of energy and feel like smiling. In teaching I can’t predict when things are going to go well and when they’re going to go poorly, I can’t predict what will exhaust and frustrate me, what will bring me down, what I just let slide off my back, or what will make happy.

When Things Go South

I’m teaching the basics of exponential growth to my fourth hour Algebra 2 class today. We’re using guided notes that I borrowed from another Algebra 2 teacher, which is new for this class; they usually take their own notes. Partway through I notice that everyone in the class looks extremely bored. Normally this class is talkative, and today they are quiet and looking anywhere except at me or their notes. I wonder if the guided notes have made things too easy for them. My goal was that they would be able to listen better if they didn’t have to spend so much time copying down notes. In the moment I notice their great disinterest, I start to panic inside. I hear myself talking and it sounds flat. All the energy is gone out of the room. How do I get it back???!? This is something that terrifies me in the classroom. It’s happened before and I haven’t figured out a fail proof way to handle it.  So I just keep talking. I feel like stopping and almost do for a moment, but what would I do then? After a minute, I remember that I had a video to show if there was time, so maybe I should just skip to that. As I’m getting ready to switch to it, I say to the class, I don’t know if you guys are bored or this is too easy or what. A couple of people mumbled that they were bored. Okay…. Was it me? Or their mood today? I don’t know.

So I go ahead and get the projector running and show a clip from Futurama about Fry checking his bank account. (See Dan Meyer's Three-Act called Fry's Bank).

Before Fry was transported in time he had 93¢ in his account. It had an interest rate of 2.25%. When Fry heard how much interest his 93¢ had accumulated over 1000 years (it was beeped out for the students so they could to figure it out), he started drooling and dropped on the floor. The students calculated the amount pretty quickly using a basic exponential function A(t) = a(1 + r)^t. (We haven't done compounded interest yet.) Then one student asked how much Fry would have in his bank account if he had started with $100 instead of 93¢, so we worked that out. We talked a little about how interest rates might change over time, the effects of inflation, and how much Bill Gates would have in 1000 years at a yearly interest rate of 2.25%. I enjoyed these slight tangents. After that we did another problem, and then I gave them 5 minutes to work on homework. I also gave them an exit slip and I was really surprised at the results – almost everyone felt it was easy!

I don’t know what the moral of the story is… Maybe to have something interesting prepared in case things go south? I was just glad we got through it and I was able to get the students’ attention back. A little confidence-booster for the future.

Algebra 2 Test and Reflections on the Last Chapter

I gave a test on Thursday this week in Algebra II. My fourth hour class didn’t do so great. I wonder if some of it was the test – I weighted the problems so that ones with more steps are worth more points, and I’m not sure if that benefits students or not. But mostly I think I need to provide more structure in that class. (My seventh hour class did much better on the same test.) I thought almost all afternoon and night on Thursday about how to deal with their poor test grades. I decided to let them correct the tests and give them 25% back. I also wanted to have a conversation with them about how we can improve their learning. I offered up things I am going to change: homework due the next day, going over homework, and notebook quizzes. I didn’t tell them that I am thinking of changing the way I present material to them. I also them what I would like to see from them: taking notes, paying attention in class, doing homework, asking questions, and using resources like me, Mr. Turner, their book, and the internet. I was planning on going more in depth in this conversation by having them think about what I could do to improve and sharing it, as well as writing down that they could do to improve their learning for the next chapter. They didn’t seem to need much more discussion, though, so I only gave them my little spiel. Then I offered the test corrections and they seemed satisfied with that. I spent way too much time re-grading so I will have to work that out for the future, but I think some of them learned more material from doing the corrections, so I like how that worked.

I am concerned about how much time it’s going to take me to rethink how I teach this class. How to set up the homework and quiz system. How to provide more structure in my presentations. How to present topics more directly and at their level. I am also looking forward to trying something this new system. I had been kind of going along at the same pace for a while, and I’m glad to have something new to try. I feel like there are still a lot of things I missed that I could have learned from teaching this chapter and from how the test went. I have a long ways to go, but I am slowly learning what students need to help them succeed.

Classroom Management / Discipline Success!

Teaching success today!
One little thing to share...
One of the Algebra I classes is a little squirrely, as they say out here in Indiana. Yesterday my cooperating teacher really yelled at them, and today she pulled a student out and took him to the library. So I'm in there by myself teaching a lesson on quadratics. I have a wireless mouse that I'm using for the first time in this class, and it's on a desk at the front of the room with the rest of my stuff. I am writing something on the board and turn around to click the mouse to change slides. Lo and behold, the mouse is missing. I know it's one of three kids on one side of the room, so I go over there and ask for the mouse back. No one offers it up, and I don't know who took it. I don't want to stare only one of them down because then they'll be offended that I think it's them, especially if it isn't (though I'm sure they were all in on it). After about 30 seconds, one of the kids pulls it up from where he was hiding it under his desk. I'm a little mad at this point because I think it's childish and I want them to be paying attention to what I'm saying, not pulling pranks. I take the mouse from him, and suddenly I think to say - "Why don't you take the mouse and click it to change slides for me?" So I give him the mouse back and ask him to change slides. The rest of the class he is mostly well-behaved, and he doesn't put his head down as usual. It was a miracle! I never think of things like this on the spot, and today I did, and it worked! Now I'm thinking that in the future I might have him or other students control the mouse for me, as long as they do it responsibly. It might make them feel more useful, responsible, and interested.

On a side note, this same cooperating teacher uses sentence-writing as a means of discipline. She has three consequences for students who misbehave in class -  a warning, writing sentences, or going to the vice principal's. I always hated the idea of making kids write sentences. It seems like such an old-fashioned and ineffective punishment. However, it seems to work for her because she's been doing it for many years and she runs a really tight class. On the way home from school today, my carpooling buddy suggested if I want to use sentences, I should have kids write affirmations. So instead of having them write, "I will be respectful and not talk when Ms. Fero is talking," you could have them write, "I am respectful and will talk only when called on in class," or "I am respectful and will wait until others are done speaking to talk." I feel like there's a negative affect on students when you have them write sentences about their negative behavior, and I really like the idea of affirmations instead. My carpool buddy also suggested that writing sentences might help kids calm down if you treat it as a time out instead of as a punishment.

That's all for today! Life is getting extremely busy here.