Friday, October 21, 2011

A Move-Around Friday Algebra Activity

I just had a really fun (and mostly productive) Friday afternoon with my Grade 7's! :) We had been working on drawing and solving some simple algebra balances, and I wanted to give them a bit more practice without it being too boring on a Friday afternoon.

This is how I structured it:
* Each kid gets a strip of one equation. They take a sheet of scrap paper and draw the balance scale that corresponds to that equation (They can ask me for help, of course, but NOT for me to see if it's correct).

* They go tape their algebra scales up around the room. (The kids were very silly about this; they taped them on all kinds of surfaces like the fan, the clock, and our lights.) The equation is written above the picture in nice, big letters with markers.

* They go stand in front of a scale that's not theirs and double-check the balance scale that has been drawn, fixing it if necessary. (I should have had them rotate once more to get a second look, but all of the mistakes except for 1 were caught.) When/if in doubt, I went over and took a look myself before they made any changes to the already-drawn scales.

* I handed out an answer sheet with the equations written on it. They are to roam around the room and find the balances that have already been drawn, and then solve them. Some of them are easier to solve than others; some of them required the kids to copy down the balance in order to work it out. (Each equation had a number/index, to make it easy for us to go over the answers in the end.)

* Since some of the kids were absent today, we needed to do two rounds of this in order to finish all the equations. The second time they sat down to draw the balances, they worked in pairs to make sure their drawings were definitely correct.

In the end, I went over the answers and asked the kids if they had questions, and everyone said that they understand the concept (but made some arithmetic mistakes)!! :) :) SCORE. It is a shame that some of my kids were absent today because of a soccer tournament, so next week I'll still have to think of a way to help those kids bridge their understanding gap. But, I think today was a very smooth run!!

Here is the material, if you wish to do this. Notice that I intentionally included parentheses into every problem; I really wanted kids to be able to see 5(x + 2) as drawing x and +2 each five times. I also threw in some "no solutions" and "all solutions" ones in there; the kids particularly liked those. :)

Wednesday, October 19, 2011

An Excellent Resource Book

I wrote about this resource a while ago, but I wanted to come back to it since I've just had a chance to scratch the surface of looking at this resource book and doing its problems. The book I bought called Graphic Algebra is absolutely fantastic! I plan to use it with my most advanced 8th-graders as a tool to encourage them to think critically about algebra and to gain familiarity with the graphing calculator at their own pace. From now on, any time it does not make sense for me to start a new skill or topic with them, while the others in the class still need some more time to wrap up their practice, I am going to point them at pages xeroxed from this book. I think it's better this way, because this book is systematic and intentional in the way that it sequences its tasks, which my onesie, twosie on-the-spot differentiation tasks cannot accomplish.

I sat down and made the answer key for the first 7 pages (involving linear functions only), and noticed that even the equations aren't totally easy to write (a good challenge for those higher level kids). Once they put the functions into the calculator, in order to navigate to the answers, they will still need to figure out how to zoom, trace, adjust table settings, and scroll through the table of values on the graphing calculator.

It's really wonderful! The topics in the book actually go hand-in-hand with what we're learning in class. First the book goes through some analysis of linear functions and related linear functions, and then immediately after, the second topic in the book is going to be linear regression.

I AM SO HAPPY!! It is actually excellent (and well-scaffolded*) material that I don't have to modify before giving to the kids. What a gem!

*Previously, when I was skimming through the book, I had thought the scaffolding was a bit too much. But now that I actually sat down to do the problems, I don't think so. That opinion could change when I move further into the book, but so far I like what I see.

Tuesday, October 18, 2011

Challenge of Pacing a Bimodal Group

I've been thinking about ways to address the bimodal performance that my Grade 8 kids exhibited on the last exam.

Good news: I am getting a lot of motivated kids to come see me during lunch for extra reinforcement to fill in the gaps of their learning.

Bad news: I think the bimodal performance is a result of my not very effective attempt to keep slower and faster learners paced similarly in the same class.

Imagine a typical class: I give kids handouts to work through at their own paces. The problems are scaffolded, so they get harder as you go. Kids are helping each other through the assignment, although there is a healthy amount of struggling happening per kid, and that means that each kid is moving at a different pace. The class works steadily for the entire period, and at the end of class, some of the faster kids finish the worksheet and grab a new one from me, while the slower kids are about two thirds of the way through with the worksheet. I consider assigning the last third as homework, but I also consider the fact that it's the hardest part of the worksheet, so I change my mind and tell the class that they can continue working on it the next day.

Fast-forward to the next class, or two classes from now. The faster kids are now a couple of worksheets ahead of the slowest kids. Instead of giving them pure algebra practice, we now are attacking the problems from all angles -- word problems with non-trivial context. My hopes are that this way, all kids are benefiting from the new assignments (whether it is additional algebra practice OR additional exposure to new contexts). I give the most advanced kids an extra challenging assignment, and I decide that it's definitely time for all of us to sync back up on our pacing, so I assign close to an entire worksheet as homework for the slower kids. The slower kids feel punished for working more slowly than their peers, even though I explain that we need to get everyone more or less on the same page now.

In the end, my slower-working kids are the ones who should probably be doing extra problems just to keep up with the material, but the reality is that they need to spend extra time at home just to finish the regular assignments that everyone else can finish in class. Naturally, their learning results are not where I want them to be, because my pacing is at least somewhat dictated by the fact that half of the class is absolutely mastering quadratic operations forwards and backwards.

How do I differentiate my instruction enough to fix this??

Sunday, October 16, 2011

IB Portfolio Task Preparation

I just spent an entire afternoon trying to wrap my mind around the IB Portfolio Type II Tasks. I found a fantastic podcast on Youtube that I am going to share with my kids: Part 1 and Part 2. What I really liked about this podcast is that the teachers walked through the technical aspect of the graphing program, as well as reviewed the rubric in parts relating to their thinking-out-loud about the sample task. I am going to assign as homework these two videos for my kids to watch at home, and then together I will go over it with them again in class, pausing every so often to go over the mathematical content and to allow them to install/run Autograph with my assistance. (There are some things that the video glosses over that the kids might find tricky to navigate on their own.)

It seems to me like many Type II tasks want the kids to "analytically" come up with their own equation, which can mean plugging in multiple points and solving the system of equations in order to find the coefficients, OR using what they know about the meanings of the coefficients (ie. amplitude or frequency) in order to construct the equation algebraically from the graph. Then, the tasks call for the students to either modify their equation to match additional data or ask them to generate (using technology) a different type of regression equation. The latter isn't always trivial to do, especially since sometimes the task gives them a fairly hairy form of equation to play with. And then, assuming that they can successfully do the mathematical modeling, they'll need to firmly link the asymptotic behavior of their regressed equation to the context of the problem, in order to examine whether that asymptotic behavior makes sense or if the modeling domain should be restricted only to the given set of data. In some cases, the kids may even need to have some background knowledge of the topic in order to properly discuss the asymptotic behavior.

In order for the kids to be successful, we'll need: 1 day of going through the basics of regression and rehearsing the technical aspects of the graphing software; at least 1 day of looking at different types of non-perfect numeric patterns, in order to figure out what type of regression is necessary (Some types are far messier than others; the IB tasks aren't messing around with simple quadratic or even cubic regression. It looks like I'll have to go into some fairly involving functions review with my batch of Grade 12's...); another 0.5 day of looking at asymptotic behavior.

Even though the task is going to surely be challenging, I think it's going to be really good for their mathematical minds to tackle this task. As a teacher, I really like how the IB Program has high expectations for all kids, because those are some very noble and worthwhile goals for us to shoot for in the math class!

(More to come about Type I tasks a few months from now...)

Saturday, October 15, 2011

Algebra Scales Worksheets and Longer-Term Grade 7 Vision

I've been taking a pretty holistic, integrated approach to my Grade 7 curriculum. Our last test looked something like this and it covered some mental percentage arithmetic, some patterns and writing of equations, and then some basic word problems. The class did fair on the exam; most kids got most things correct, except for the setting up of the (rather complicated) word problem. It showed me that we're going to have to come back to practice that skill before the next go-around on an exam, but that as a class we're ready to move on to a new topic -- proportions, but in an integrated fashion still, while looping in all the things we already know.

So far, in terms of actually solving equations, the kids have not progressed to formal symbols and algebra yet. Since I promised them that every Friday is going to be something "fun", I made up yesterday a sheet of algebra scales problems for them to do. (Normally, I'd do this kind of thing on the computer, but for reasons that are not easy to explain, I couldn't get easy access to the computers at the school. I figured doing it on paper is almost as good.) Here was the file I used, and the kids were a bit nuts about it while working loudly but enthusiastically in groups! The problems are scaffolded up to letting the kids see when there are no solutions or there are infinite solutions, and then on the back side they needed to draw their own scales from a given equation, in order to solve for x. I used different levels of the scale to show why we might have something like 5x + 3x + 2 = ... and why that's equivalent to 8x + 2 = ... In a few weeks, once the kids have internalized this visualization method (including negative coefficients and negative integers), we'll go over how it translates formally to algebraic symbols. As always, I am a firm believer that symbols need to be introduced only after the visualization of the operations becomes second-nature to the kids.

It's probably nothing new, but I think if you don't already have a worksheet like this, you may find my scaffolding helpful, so here it is: Positive things on scales and Negative things on scales.

Isn't algebra fun?? :) My hope is that by the end of the first semester, my kids will: be comfortable with the idea of predicting linear patterns forwards/backwards and reading/making graphs; understand intuitively what proportions are, when to use them, and how they are tied to linear patterns; have great number sense and be able to quickly compare quantities that involve calculating fractions or percents; be able to set up and solve linear equations given a word problem. I think the list is ambitious, but definitely doable. That would leave us time in the second semester to do some probability/basic geometry and to begin tackling a "harder" algebra topic such as quadratics.

Thursday, October 13, 2011

Integrity Brain Dump

I've been reflecting extensively about integrity, because this year I am really trying to do a better job with the character education program I am teaching as part of the Grade 8 homeroom. Already I've talked to the homeroom kids about: appreciation as a tool to smoothing over difficult confrontations; manners and cultures; why we come to school (and why that's a privilege); the idea that our intelligence is not fixed, and that our knowledge can improve our brain's processes and in effect we can "get smarter" over time; setting goals and brainstorming study skills. I have found that by opening my own life experiences up to the kids, I am making the textbook lessons more meaningful for me (and thereby, hopefully for them as well). Every week, I try to tie the lesson to something a little bit deeper, and the kids have reciprocated by opening up with their own thoughts and experiences as well, allowing me a sneak peek at their thoughtful and insightful side, which I don't always see in the context of a math class.

Something that has been on my mind (although not necessarily in the Grade 8 character-development curriculum) has been the issue of integrity. The more I think about it, the more integrity seems to weave itself into everything that we do, which means both that it's necessary for me to talk about integrity issues with the kids, and also that it is very difficult to get a clear, convincing, effective message across in a single discussion. I am going to do a brain dump over here; feel free to add your own thoughts.


When we think about integrity, we mostly think of not lying and not cheating. In school, we only talk about integrity as a reactionary mechanism -- usually after something bad has already occurred, such as group cheating or plagiarism. What we need to do is to actively address integrity issues as a school as a prevention mechanism -- and that can be done in our individual curricula (ie. history and English, where morality is explored in the context of literature), or in our character-education classes.

The first stumbling block I have is how to talk to kids in a convincing way about the importance of integrity. Why should they bother not doing certain things, if they know that they cannot or probably will not be caught?? I think we are dissuaded from negative behavior for any number of these reasons below (depending on our own morality developmental stage):

* Being worried about consequences (ie. punishment) that may occur to us personally
* Upholding our reputation (another form of personal consequence)
* Being afraid to hurt or damage others who trust us
* Believing that our individual needs should come after our commitment or obligation to a group/policy/community (Relativistic morals?)
* Believing that the action is wrong on an absolute scale

As you probably agree, our aim to talk to kids about integrity is roughly equivalent to moving them along the spectrum of reasons towards the intrinsic motivators rather than the extrinsic motivators. But, doing so is difficult. The best that I can come up with is talking to kids about my own view of my own personal integrity, in order to shed light on what it means to me, personally. (I grew up in a family that raised us on stories that carried inherent values, but I'm not sure if my kids have the same relationship with their parents.)

So, here are some examples I've come up with for where I think I exhibit a personal integrity, in a way that is perhaps subtler than not cheating and not lying:

My integrity is reflected in the way I work at my jobs. When I was 17, I had my first job evaluation by my Starbucks store manager. I was nervous, and I had asked my friend if he was nervous as well. This is what he said: "I always do my best on a job, whether or not someone is watching me. And in the end, that's all I can do. There is no reason to worry." I still carry that work ethic with me today. I don't ever compare myself with other teachers in my department; I need to do my best within my own frame of possibilities. Over time, that frame will expand. So, it doesn't matter if someone is years more experienced than me or if they choose to leave at 3pm; what matters to me is that I do my very best every single day for every kid in my class, within my own realm of possibilities. Furthermore, I don't ever worry about whether I've earned my place at work, because I've never lied or cheated on any test, project, resume, or interview to get there.

My personal integrity also means to me that if there is a legitimate way to get something done, that's what I'll do even if that means my life will probably be made a bit harder. I remembered today an incident where I went to a supervisor asking for a day off after months of not taking a single day off -- there had been days when I was so sick that I could not stand up, and still I had shown up to work. The supervisor told me no, that I couldn't have this day off to go to my friend's wedding, even though I was requesting it far in advance. I was really upset, but even then I refused to call in sick that day. My personal integrity means that I need to represent what is true, even if that truth would inconvenience me.

When we put ourselves in the shoes of a hiring manager, it's clear why integrity is important. Whom would you rather hire -- a conniving employee who might lie about their results, or an employee who would own up to their mistakes and reflect upon them with others? For the same reasons, we prefer our politicians to have integrity, because so much goes on behind closed doors in politics that we have to be able to trust them (at least a little). When the situation is gray legally but not ethically so, we can only hope collectively that those in charge (such as the bankers giving out mortagages) are doing their part to ensure that the interest of the larger community is protected. Integrity, therefore, is clearly something that we value as a community.

When does a kid encounter integrity issues? All the time, I bet. Inside and outside of school, I bet. Imagine a kid whose parents want him to go home early, while the temptation is to hang out late with his friends. He can either: 1. stay out and then make up an excuse afterwards, 2. go home and then sneak out, 3. try to reason with his parents to get a later curfew, but going home at the promised time, even if it's early. Which is the least pleasant but the most honest option for a kid? Probably #3. In situations big and small, whether or not they even give it a second thought, they are constantly being confronted with choices and reinforcing their own integrity, or the lack thereof.

So far, these are all just my thoughts. I am giving an anonymous survey next week in my homeroom and I'll be tallying the opinions to get a feel for what kids consider to be cheating behavior, whether that definition is tied to the outcome (ie. if they successfully cheated or not), and what they think are the most important factors that discourage them from certain types of behavior. From those surveyed results I'll plan some discussion points and go about it, leaving much of it open-ended to hear what the kids have to say about the issue.

The point is that I want to open the door for more proactive conversation about integrity and less reactive conversation. Thoughts? Ideas??

Addendum 10/14/2011: In case you are interested, here is the survey I am going to pass out.

Wednesday, October 12, 2011

Bimodal Performance

One of my classes just took an exam and exhibited bimodal performance. Half of the kids did very well and did not miss any conceptual understanding, only a very few (1 or 2) arithmetic errors throughout the entire exam. The other half of the class is missing large chunks of the conceptual setting-up-of-equations and making-predictions type of work. The split is about even.

So, what to do now? My intuition is to complete a concept map/graphic organizer as a class and then set them up in expert-novice pairs to work on remediation, and then the kids who need the extra practice are going to go home with a fresh sheet of problems and be asked to work through it to get additional reinforcement.

What would you do?

PS. I think one of my mistakes was to wait too long to have a large assessment. That's obviously easy to fix in the future, but I'm curious what you think I should do now, given the current situation.

Monday, October 3, 2011

Shanghai Exchange

I just spent a weekend hosting four teachers from a traditional public middle school in Shanghai. Our school has an exchange program with them, where they take a small group of their students here to visit in the fall, and we take a small group of our students over there to visit in the spring. The kids stay with host families and go with their host students to classes, and the adults are hosted by local adults in order to get an authentic feel of the place.

It turned out to be very useful that I can speak Chinese. Two of the teachers spoke English very well, as they are English teachers at the school. A third one understood a decent amount, but a fourth one did not speak any English at all. It was not necessary, therefore, for me to speak Chinese, but the fact that I could helped all of them feel comfortable. Geoff tagged along to also be a tour guide, and we took the teachers to see all the usual touristy places.

We started off in Alexanderplatz on Saturday, took some pictures at the Rotes Rathaus (Red City Hall), and then they declined paying to go up the TV Tower, which is the highest point in Europe. I then took them shopping (as per their request), and after lunch we went over to the beautiful Reichstag, which is where the German laws are made. We had made a reservation beforehand, so we got to go up to the beautiful glass dome of the Reichstag, and to take a self-guided audio tour that introduced us to the features of the buildings surrounding the Reichstag and the features of the dome itself.

On Sunday, we first went to Schloss Charlottenberg, which is a beautiful palace built hundreds of years ago during the Prussian dynasty, and still today reflects the luxury of those times. During WWII, much of the palace was bombed and destroyed, so much of what you can see today is the result of reconstruction in the 80s. Still, you can get a sense of the grandeur that once dominated this palace. After some hours at the palace, we headed over to the famous Checkpoint Charlie to take some photos, and then we walked along the Eastside Gallery, which is a stretch of the remnants of the Berlin Wall that has since been turned into a symbol for hope and inspiration as artists have made the wall into an elaborate art display.

During the course of the day, I got a chance to ask about the school in Shanghai. One of the girls told me that in Shanghai, there are four tiers of schools: city-level magnet schools, district-level magnet schools, "normal" schools, and private or independent schools. At the end of every level of schooling (ie. elementary school, or middle school, or high school), kids need to take a city-wide test and apply for the next schools. The system is very competitive, because in order to get into a good college (or perhaps any college at all), you need to be from a top high school, which means you needed to be from a top middle school. There are some exceptions to this system, however, such as the fact that a kid who lives within a certain close proximity to a school has the right to attend that school, even if the kid is not academically qualified. And, on top of that, there is a lot of pressure from the government to make sure that ALL kids pass every class by the end of middle school, regardless of whether the child was qualified to attend this school in the first place. So, that creates a lot of pressure on the teachers AND on the kids who need to struggle to pass just ONE class, let alone all classes.

In Shanghai, this teacher tells me that they teach roughly only half of the time that I teach, but that in every class they have 40 kids. I asked her if she thinks she has enough time to reach every kid and to take care of them, and she said no. Most of her prep time is spent on correcting the daily homework that was assigned. The kids go home and most of them do their homework through midnight each night. So, it is an eye-opening experience for their kids to come to our school and see that our middle-schoolers have barely any homework and enjoy so much freedom at home and in school.

I look forward to visiting their school in April to see for myself what it's like!