Wednesday, September 22, 2010

Days 16 - 19

Sorry, I got kinda busy the last couple of weeks, so I’m going to try to cover all of last week in this post (obviously not quite as in-depth as usual), and then try to catch up with individual posts for the four days this week. To save time I’m going to simply start linking to the individual days on the class blog where you can find links to the openers, lessons and videos, instead of linking them individually. I hope that works for you. Here we go.

Day 16
Today we took the assessment for Solving Equations with Variables on Both Sides. I was concerned about giving an assessment on a Monday for all the usual reasons, but with a four-day-a-week class and the frequency of my assessments, there’s really no good way around it. (Plus, philosophically, I don’t think it should make a difference giving an assessment on a Monday, it they know, they know it.) The results were pretty much what I expected, which was slightly disappointing. They didn’t do as well as I had hoped and it was clear that many of the students hadn’t thought about Algebra since they walked out the door at 8:20 am on Friday. I’m really struggling with the lack of effort some of the students are giving outside of class (and this seems to be a common complaint among other teachers of freshmen at my school this year).

After the assessment we did a quick review of graphing an equation by making a table, and then moved on to the lesson, which was a distance/rate/time problem that we attacked using recursive sequences. Once again I’m worried that I’m scaffolding this too much for them, but whenever I try to back off and let them take more ownership they flounder. This is going to be a continuing theme for me I think. My other concern today was their continued difficulties with the concept of speed and how you figure out how fast someone is going. We have already done a few distance/rate/time problems and I know they’ve done some in science before, plus I thought that speed was somewhat of an intuitive concept for most students. Yet they can’t seem to translate “how far I’ve gone and how long it took me” into actually calculate a rate.

Their homework was to watch the Graphing Linear Equations by Using a Table video as well as finish the lesson for the day. This was a little more than I intended to give them, but we didn’t get as far in class as I had hoped, and they do have two days to work on this. I encouraged them to come in on Tuesday (the day they don’t have me) for help if they were struggling.

Day 17
This was a day I had high hopes for in terms of engagement, as I invited in several folks (two assistant principals and our media specialist) to be guest participants in an activity. We walked down to our gym hallway where there was a little more room, I setup a 50 m tape and then the guests performed eight ten-second “walks” as the kids noted their position at each second. I had given the participants written directions ahead of time (“Start at 0. Walk at about 1 m/s for 3 seconds, then stop for 3 seconds, then walk at about 1 m/s for 4 more seconds”) and then I counted out the seconds and the students wrote down the positions.

This turned out to be more difficult for the students than I had expected, and then collating the data took a long time (which I did expect), so we’re going to finish this activity on Friday.

Their homework was to take the Graphing Linear Equations by Using a Table pre-assessment and to make sure they had all the data we collected written down.

Day 18
This was another Carnegie Hall day (“How do you get to Carnegie Hall? Practice, practice, practice) to give them some repetition of some skills and hopefully prepare them well for their assessment tomorrow. My class seems all over the place on their ability to plot points (something they’ve done before in multiple classes), and they are still struggling mightily with simply substituting in values and evaluating expressions correctly. I told them today that I’d love for them to be able to do this without a calculator but, if they find themselves getting things wrong often, I want them to use the calculator to be sure. So, do it in your head first, then verify with the calculator. I don’t think I convinced many of them.

Their homework tonight was to finish the review worksheet if they wanted to (we did most of the graphing problems in class, but then there was a spiral review of earlier topics), or they could save the non-graphing review problems for the weekend if they wanted to. Then they also needed to review for the assessment tomorrow. I feel like I’m communicating well in terms of what they should do to review for assessments, but I’m not sure I’m getting through.

Day 19
The assessment over Graphing Linear Equations by Using a Table was today – and, overall, they bombed it. Several factors here I think: Homecoming Week, two assessments in one week, me not standing over their shoulder making them do the homework, their difficulties in substituting and evaluating, and perhaps too high of expectations on my part in terms of their readiness to take ownership over their own learning. Back to the drawing board.

After the assessment we then worked with the data we collected on Wednesday and they seemed to be getting the hang of it. We had a shortened class due to the Homecoming Pep Assembly, so I once again left them with more of the lesson to finish for homework that I would like (this is getting to be a bad habit on my part – I really need to fix my timing issues). They also need to watch the next video, Graphing Linear Equations by Using Intercepts over the weekend.

I felt like this week was a step back for me and my class, hopefully next week will be better.

Saturday, September 11, 2010

Day 15

Today went well (I think). It felt much more relaxed, perhaps because I got the timing right for one of the few times this year. The students have also seemed more relaxed the last few days, joking around with me more at the beginning of class (although they still get too quiet as we move deeper into class and content).

The opener (pdf) was designed to reinforce some basic distributive property stuff, give them some practice on solving equations with variables on both sides (formal assessment over that is on Monday), and to remind them/introduce them to the concept of a sequence. They did reasonably well with these, although I’m still concerned with the number of students who can’t do problems like 1a correctly (with a calculator).

The lesson (pdf) was then an introduction to recursive sequences (this is leading to writing linear equations and the concept of slope). It was also an opportunity to show them one of the features of their graphing calculators – the ANS key. (Not all of the students have graphing calculators, but I made new groups this week to make sure at least one student in each group, and usually more, has a graphing calculator).

So they built the patterns out of toothpicks and pretty easily saw the rules for number of toothpicks and perimeter. I then showed them how they could build a recursive routine into their calculators using braces and the ANS key so that they could quickly generate results for additional figures. They then pretty easily came up with the number of toothpicks and the perimeter for figure 25. (As cool as this is, we’ll eventually transition to, “Well, what if we needed to know figure 125, or 1125? Wouldn’t it be nice to have a quicker way?” Bingo, let’s right an expression/equation.)

They fairly easily then replicated this process with a sequence of squares, although I did lose a few students who were more interested in creating artwork with the toothpicks. My favorite quote of the week was from a student who said, “Mr. Fisch, do you ever worry that parents are going to call you when they find out we were playing with toothpicks in Algebra class?”

Their homework was to prepare for their assessment over Solving Equations with Variables on Both Sides on Monday (and, yes, I’m a little worried about giving it on a Monday), and to plan how they might want to participate in Homecoming Week (pdf) next week.

Thursday, September 9, 2010

Day 14

Today went better. It was a Carnegie Hall day.
How do you get to Carnegie Hall?
Practice, practice, practice.
They seemed more confident on the opener (pdf), even though questions 3 and 4 were something they just had last night in the video that was for homework. Now, that doesn’t mean they all understood how to do it, but enough of them did that it makes me think we might be on the right track. Then, as the students worked them out on the smart board and I then went back over them, I saw several more light bulbs go on for students. (And, later in the day, I had a student come in for help for the first time unprompted. I’ve had students come in before, but always prompted directly by me.)

We then attacked a series of problems (lesson, pdf), divided into distributive property (they did well on) and solving equations with variables on both sides (mixed results, but definite progress). I had hoped we would get through the third section (graphing on a coordinate plane), which would’ve just left review problems for homework, but no such luck. Still, they only had those two sections for homework, as well as completing their online pre-assessment for solving equations with variables on both sides.

I had this day planned well in advance but, in retrospect, it would’ve been better if I’d done this yesterday. I think the students would’ve been much less stressed, and I know I would’ve felt better about things. If I’m still in the classroom next year, this post will help remind me what to do differently.

If you’re kinda sorta following along with this blog, take a look at Jason’s thoughtful comment on flow on yesterday’s post. He makes some great points, and in my reply comments to him I try to explain why I’ve made some of the choices I’ve made. I appreciate the comments folks are submitting, as it’s helping me re-examine what I’m doing from another perspective and sometimes make changes (and sometimes re-justify to myself why I’m sticking with something).

Wednesday, September 8, 2010

Day 13

Today was . . . confusing. I’m pretty sure it was confusing for my students, but it was also confusing for me. We started with the usual opener (pdf) and they seemed pretty stumped again. At this point I’m expecting them to be able to do a straightforward order of operations problem, particularly because they have calculators. That wasn’t the case, as some of them struggled with 1a.

I knew they would struggle some with 1b, as I could tell when they did the openers on Friday that they weren’t very comfortable with distributive property. (Note to self: I need to find out for sure what they’ve had in middle school. I know in the past that distributive property was something they had been exposed too, but now I’m wondering if perhaps that’s not the case anymore). I wasn’t that concerned that they struggled with 1b, as this was another opportunity for them to see how to do it, and because I know tomorrow I’m going to give them more practice with it.

Question 2, however, was a little bit depressing. While I knew some students would struggle with it, I wasn’t prepared for the number of students who apparently had no idea where to start. I referred back to what we did on Friday (suggesting they look back at our work from Friday if they didn’t remember), and referred to the video they watched on solving two-step equations. I was anticipating that they would be able to do column 3, “undo the steps,” and struggle a bit with column 4, “The Algebra,” yet many were unable to even start the undo the steps column. I think this confirms my supposition that at the end of the day Friday they didn’t get it quite as well as I had hoped, but I’m also frustrated since I thought the video they watched should’ve helped solidify the idea of simply doing the inverse (opposite) operation. When I ask how to “undo” something they all seem to know, yet they can’t come up with what to ask on their own. I’m hopeful that when we do some more practice problems tomorrow (no new stuff) it will begin to click for more students.

Then the lesson (pdf) today was equally frustrating. (Note: image from Discovering Algebra from Key Curriculum Press). We did what I thought was going to be a quick and straightforward experiment, using their textbooks as a ramp and rolling pencils off of them, and then measuring how far they rolled. Then we were going to plot the points and see if there was a relationship (greater the height of the ramp, the farther the pencil rolled). I’m not sure if it’s just too early in the morning, or if I did a really bad job explaining this, but they moved through this so lethargically that it felt like by the end they had no idea why we did it. Again, I’m feeling conflicted about how much scaffolding I need to give them to be successful, versus my philosophy that they need to be problem solvers themselves.

Then we moved on to a data set from the U.S. Department of Transportation, showing the average fuel efficiency of U.S. passenger cars by year since 1980. I pulled the chart from the web and then started a table for them. I then asked them to copy the year and the mileage from the chart into the appropriate columns in the table, and then in a third column they had to calculate the years since 1980. I demonstrated with the first three points to make sure they knew what I was asking them to do, then asked them to complete the rest of the table (they did not have to copy the actual chart itself). That proved really difficult for some of them, so once again I’m questioning my assumptions about what is reasonable to expect 14 and 15 year olds to do. Should transferring information from the chart to a table, when I setup the table and help them with the first three data points, be pretty straightforward as I think it is, or is that unreasonable?

Either way, that took much longer than I anticipated (yeah, back to my poor timing issues that I thought I had resolved). As some students had finished their tables and others were still working, I revealed step 2 so that the students who had finished could start on their graphs. Again, I worried that I was scaffolding too much by telling them to scale both axes by 1’s. It turns out that that wasn’t enough scaffolding for many, as they labeled their x-axis 0, 5, 10, 11, 12, . . . Once again, my assumption was that they have done some graphing previously and that this wasn’t completely new for them. So, again, I need to find out for sure (in my own defense, I did ask the students if they have graphed previously and they say they have).

I then talked through steps 3 through 5 to make sure they knew what I was asking for, and then gave those to them for homework (as well as a few will need to finish the graph as they did not finish in class). I’ve been working really hard at not trying to play “gotcha” with checking their homework, but I think I may be wavering a little bit on that. Every day I talk about the importance of doing whatever I’m asking them to do outside of class, and the importance of coming in for help when they don’t understand, but the blank looks are starting to wear on me a bit. My philosophy of continuing to trust them and share with them the reasons I’m asking them to do things, and then giving them time to figure it out and start stepping up will eventually have to end if they don’t step up. As I said to someone I was talking about this with today, eventually if it’s not working for my students, then I’ll have to change something as I do have a limited amount of time with them (even if I think that’s detrimental to them in the long run).

Today felt like I stepped back about two weeks.

Saturday, September 4, 2010

Day 12

So today seemed to go really well until right until the end, but then it seemed like maybe they weren’t getting it after all. Of course, it was the Friday before a three-day weekend, plus we don’t meet on Tuesday so I won’t see them until Wednesday, but still.
We started with our usual opener (pdf), but I tried something different today. As I’ve mentioned before, I’ve been surprised about how long the students are taking to do what I consider to be pretty routine stuff. So, each day on the openers I’ve been saying things like “I think these should take you about five minutes” or whatever, but then they take ten minutes and I’m not quite sure what happened.

So today I decided to make it a little more explicit, I put suggested completion times after each opener. Now, I want to be clear that I don’t believe that speed is the most important thing here. I don’t want to rush my students. Yet I also believe that students must have a comfort level and facility with the mathematics to do things at a reasonably good pace if they’re ever going to be able to move forward and work on more interesting – and complex – mathematics.

So at the beginning I told the students what I was doing, and that they shouldn’t panic if it took them longer – or shorter – to do each problem. I told them that those times were the times that were the target for them, where I would like to see them get. I also told them that the total time on the timer was slightly more than the sum of the times, so that I didn’t push them too much (I think I had four minutes on the timer in the upper right corner, even though the suggested times totaled three minutes). When the timer went off, I then said that if they were finished, or working on #3, they were in pretty good shape. But if they weren’t even on #3 yet, then perhaps there was some concern there and they might want to come in and get some help. They then had several minutes to talk over the openers in their groups before we worked them on the smart board.

I’m not sure how I feel about this, as part of me really hates doing this. But they did seem a little more focused today, and I’m going to try it for a few weeks and see what happens.

We then moved on to the lesson (pdf), where we approached solving two-step equations via the idea of working “magic” with numbers. So, here’s the scenario. I told them to pick a number between 1 and 25 and write it down. I then gave them one operation at a time (“Add 8” or “multiply by 4”). After each operation, they then wrote down their result. After about five operations – and keep in mind they have calculators for this – I then revealed the magic and told them what the last number they wrote down was (“5”, or “Your original number”).

Now, I knew this was going to happen, yet it still was somehow a surprise. A fair number of students didn’t get “5” or “their original number.” So, here’s my question to you, is it unreasonable to expect that students, using a calculator and taking one basic operation at a time, couldn’t complete five operations in a row successfully? I’m really struggling with this, because I think (perhaps naively) that they should be able to and that, if they can’t, they are going to struggle mightily in Algebra. Given the fact that clearly some of them are struggling with this, I’m not sure what to do to help them. Ideas?

After translating the “magic” to the algebra to show them why the tricks worked, we approached solving two-step equations by “undoing” operations. As we were working through this I was feeling pretty good, they really seemed to be getting it and understanding the concept, even if they were still a little shaky with the fourth column (“The Algebra”). But then I turned them loose (in their groups) on the last page of the lesson, and suddenly a whole bunch of them didn’t know what to do. Not just with the fourth column, but with the other columns. I had anticipated they would struggle with the last one (writing their own given just a result), but not that they would struggle so much with the first two.

Their homework for next week (again, I won’t see them until Wednesday), is to watch the video on solving two-step equations (in addition to completing their reflection/goals assignment from Wednesday), so hopefully that will help solidify the concept of undoing and what to write for each step.

So, overall, week three felt better, but still nowhere near where I want it to be. I did much better on my timing each day, and I think I’ve scaffolded things better for my students, but I still worry that I’m doing too much of the thinking. As always (at least until the end of the year), next week is another opportunity to do better. Let’s hope I do.

Thursday, September 2, 2010

Should We Use Graphing Calculators or Computers?

First, a little background. I strongly encouraged my students to get a graphing calculator for Algebra, but it is not a requirement. Just over half of them have one right now. The math department has a partial set of graphing calculators that I can check out if they're not being used, and they also have a set of 15 laptops running Windows XP.

I'm beginning to plan the lessons where we'll start looking more closely at linear relationships, including tables of values and graphing (in addition to equations). One of the things I hope to do is intro it to them as a recursive routine and hopefully help them see the connection between the rule, the table of values, the graph, and ultimately the slope of the line. The activity I'm planning right now has them use a graphing calculator and a recursive routine to help them generate points, then they graph those points by hand and we look at the relationship between the problem, the points, and the graph.

I could definitely do this with graphing calculators. Since the students are working in groups, the students have enough graphing calculators that every group could have one, and the math department has some that I can bring in to supplement in case any group doesn't  happen to have one that day. But as I was planning on what I was going to show them on the projector/Smart Board, I found myself creating it in Geogebra (which is just a fantastic program, even though I don't know it very well yet).

I find it easier to use Geogebra than the TI Smartview that emulates the calculator on the screen, and I certainly like the clarity of the graphics much better.In Geogebra, I use the spreadsheet function to create the equivalent of lists and/or recursive routines on the graphing calculator, then use that to plot the points on the graph. I can then add colors to the graph, easily change my scales and views on the axes, and just manipulate the heck out of it.

I think initially the push toward graphing calculators was because they were relatively inexpensive (compared to computers) and portable, and therefore it was more realistic to expect students to purchase them, or for math departments to get class sets of them. Therefore I was assuming I was going to choose graphing calculators over computers for activities such as this. But now as the cost of devices has come down, and as more and more students have close-to-full-time access to some kind of computing device, I wonder if it might make sense to transition back to using computers for something like this. Aren't we now at the point where we can assume that students will most likely (at least as they get older) have access to computing devices more often then they will graphing calculators?

Certainly the idea of a recursive routine works just as well on Geogebra/a spreadsheet as it does on a graphing calculator, and perhaps better as they have to think more about the process. (On the graphing calculator we would probably use the ANS key to generate that recursive routine, which might obscure what was going on. On Geogebra, we would write the spreadsheet formula, which is more . . .ahem . . .transparent.) Geogebra is more flexible than the graphing calculator in many respects, although certainly the graphing calculator does things Geogebra does not. So I wonder if it makes more sense to try to get the laptops and have them use those one per group for activities such as this. (Also keep in mind that Geogebra is free for Mac OS X, Windows and Linux, so no cost to schools or to students if they want to install at home. It runs fine on our netbooks running Linux.)

So, this is a long-winded wind up to the point of this - what do you think? Does it make more sense to use graphing calculators, or transition to using computing devices for math learning/instruction/exploration? I'd love to hear your perspective, whether it's as a math teacher, an ed tech-type person, or some other interested reader.

Day 11

MAP Testing today, so no instruction.

I should've mentioned in yesterday's post, however, that I did give them a couple of assignments after the Skype session and our conversation yesterday.

First, I gave them a short set of review problems (pdf), since I'm worried I'm not giving them quite enough repetition. Those are due tomorrow.

Then I asked them to write their first reflection piece, as well as set some goals. I'm asking them to do this in Google Docs (in our Google Apps installation), so I gave them a brief set of instructions in case they weren't comfortable finding it. I created the document for them (in shared folders that I've created between each student and me) that contains the prompt, they just need to open it and type in it. I've previously asked them to login and check their email, so theoretically at this point they all know how to access it - we'll see. I've told them to check before the weekend to be sure they can get to it, otherwise they should see me for help on that.

Here's the prompt:
Looking back at our first couple of weeks in Algebra, how are you feeling? What’s going well or you are excited about? What’s challenging or are you concerned about? Please answer in complete, thoughtful sentences.

Then I want you to set three goals for yourself for this semester.
  • One goal specifically related to Algebra
  • One goal related to AHS in general (can be related to classwork, sports, activities or something else at AHS)
  • One goal outside of AHS
Make these goals fairly specific, not just “I want to get a good grade.” For each one, answer with what, why and how what is your goal, why is it your goal, and how will you accomplish it.

I’ll be asking you to revisit these goals toward the end of the semester and evaluate how well you’re doing on them, so make them be good. Please also use complete, thoughtful sentences for these.

Wednesday, September 1, 2010

Day 10 - What I Said Today

[cross-posted on The Fischbowl]

Today was our Skype session with Professor Garibaldi, and I thought that went well. After the Skype session we only had about twenty minutes left (shortened classes due to a PLC day) so I took that opportunity to talk with my class a little bit. I realized that I hadn't done a good job of conveying my thoughts and beliefs about the class, of sharing my passion, of explaining why I setup class the way I did and what I was expecting from them - and what I was hoping for them.

So here, more or less, is what I said. I'm sure it wasn't quite this smooth, as when I write I automatically correct and tweak, but this is pretty close to what I said (and definitely the spirit of what I hope I conveyed).

I wanted to talk a little bit about this class and why I’m doing the things I’m doing. Mr. Krause, one of our English teachers, is doing a project right now where his students are asking people how they define success. I answered that for several groups of students, but I wanted to talk for a minute about how I’ll decide if I’m successful with you guys in this class.

I won’t think I’m a success if you get a good grade in Algebra, although I certainly hope you do and I’m going to try really hard to help you do that. I won’t think I’m a success if you score well on tests like CSAP or ACT, although I hope you do, and even though a lot of well-intentioned people think that’s how I should define success. I won’t even think I’m a success if you go to a good college and then get a good job, although I certainly want you to do that because I’d like to retire someday and I need you all to have good jobs to support me.

No, I’ll consider myself successful if you turn out to be good, kind, caring adults. If you’re a good spouse, child and parent. If you contribute to the world and to your community and help those around you. If you participate. And learn.

And here’s the deal. The education that I received was a pretty good one. But it’s not good enough for you guys. Not anymore. You see, in a rapidly changing, information abundant world, the people who are going to be successful – both professionally and personally – are the learners. And by “learners” I don’t mean people who just learn what we teach you here at AHS.

Now, I want to be clear, that doesn’t mean I don’t think you should learn what we teach you here at AHS. I don’t want you to go to your second period teacher, raise your hand, and say, “Mr. Fisch said I don’t need to learn what you’re teaching.” Please, don’t do that. That’s not at all what I’m saying. Your teachers here work very hard trying to share important, meaningful and relevant knowledge and skills. And that’s important, but it’s not enough. Because to be successful in the 21st century you’re going to have to be a learner, you’re going to have to learn how to learn, and go after things on your own. You’re going to have to be independent, curious, passionate learners, who don’t just sit back and wait for someone to tell them what they’re supposed to know, but who go out and try to figure things out for yourself. Who pursue your interests, your goals, your passions with intensity, and who actively participate in everything you do. Who go out and find other learners who are passionate about what you are passionate about and learn from them – and alongside them.

To quote myself (sorry), the world has shifted. The world of school, and the world of work, and the world in general has shifted, and so I need you to shift as well, and that’s what I’m trying to do in this class. I’m trying to get you to be actively involved in your own education, to be independent and curious learners in mathematics, even if Algebra is never going to be your favorite subject.

I believe you need the skills I’m trying to get you to learn for three main reasons. First, to be a successful citizen in the 21st century you have to be numerate. In order to deal with all the data that is going to get thrown at you, and to make good, responsible, effective decisions, you’re going to need a lot of the skills we’re learning in Algebra.

And frankly, that’s not necessarily true about all the math classes you’ll take. Honestly, if you take Trig and Pre-Calc, the skills you learn there are very important if you go into the math and sciences, but perhaps not so much day-to-day life for most of you (some folks will disagree with that). But the skills we learn in Algebra you’ll be using every day to make sense of all that data in the world, to be informed voters and decision makers.

The second reason to learn the skills is if you decide that you are passionate about math and science, you need these skills in order to progress to more complex topics and to go deeper.

The third reason – and it’s the one I think is least important but you may think is the most important – is that right now in the short term you have to learn these skills to get a good grade in this class, to do well in school, and to get into college if that’s what you choose. So while I prefer that you focus on the first two reasons, this one is still a valid one for many of you.

And this is why it’s critical you do the assignments I’m asking you to do, like watching the videos I’ve created for you. Those videos are designed to help you master the skills, and to become more independent learners. But they’re also designed to free up class time so that we can become more curious, active learners, in class, and so we can explore interesting (or not for some of you) applications of Algebra like the bike gear ratios or Tim Tebow’s speed at the NFL Combine or a variety of other activities we’ll be doing this year. In order to apply the skills in class, I need you to do the necessary work outside of class.

But in order for that to happen, in order for us to use our class time to be the kind of learners I think you need to be to be successful in this century, your century, I need you to step up and take care of business. I need you to watch the videos, and use them as they’re intended, and do the other things I ask you to do outside of class. And I really, really need you to participate in class, to be active learners. To ask questions, and be involved, and talk to each other, and help each other, and be willing to take risks in order to learn more, even if that makes you a little nervous or uncomfortable. I need you to do more of the talking in class, and me to do less. I need you to do more of the thinking, and the questioning, and the figuring out.

So I’m asking you to please, please consider what kind of future you want, not just for yourself, but for those around you, and make an effort to be as independent, as curious, as responsible, as passionate of a learner that you can be. And I promise that I’ll bring the passion every day and do the very best I can to help you become that learner.