Tuesday, November 29, 2011

My Favorite Student

My favorite student (this year) is someone who is not particularly quick at picking up math concepts. After she got the first test back, she cried. She came to see me everyday during lunch to go over a section of the test at a time. We did the corrections together, and then when she went home, she re-did the problems at home that we had worked on together earlier that day. It took us several days just to go through the entire test. And then, because she had worked so hard, I offered to give her a re-test not for grade, but just to see if she now knew the material. On the re-test, she made no conceptual mistakes, and only some arithmetic errors. When she went home and did corrections again (this time, unprompted), she thought the corrections were very easy.

This kid works so hard that when I announced that we were having a big exam in December, she came to me within a couple of days to start working on the topics that will be included. We sat down and made flashcards for strategies while approaching problems, and we did some practice problems and put them also on the flashcards. She has been reviewing the flashcards since at home, and she says that they are very helpful as a starting point to solving problems.

This favorite student of mine came to me yesterday after school to ask me to teach her long division, because all of her friends are able to divide 3 digit numbers by 2 digit numbers and she couldn't remember how to divide. I was running a fever when she came by after school yesterday and my body ached all over and I longed to go home, but hearing her question warmed my heart. Big time. She is my favorite kid because she isn't embarrassed to get help, and I know that some day she will master all of the skills that she lacks, if she can keep up that amazing spirit of hers.

That's my favorite type of student. Over the years, I have seen other students like her grow into excellent math students, who can connect the dots faster than anyone else and to work quickly through complex scenarios of problems. As a teacher, may I always remember to appreciate when a kid has the willingness for hard work, regardless of what they currently achieve.

Saturday, November 26, 2011

Introducing Limits and Derivatives

I know you non-IB people probably cannot fathom this, but Calculus is only one of many units in the IB curriculum. I am introducing it to my kids now, and it's the last big topic I plan on teaching before we start to do mixed review in the spring time.

The reason why Calculus is condensed into a single unit in the IB makes sense to me, actually. Although there are so many applications of Calculus, I would be just as happy if a kid can walk away from Calculus knowing the big ideas of limits, differentiation, and integration, and to be able to do basic differentiation/integration of polynomials by hand without a formula sheet. Everything else, I'm OK with them relying on a formula sheet in order to remember the mechanics. So, it's very do-able as a single unit and to still develop the relevant concepts as a class.

To that end, I have already introduced instantaneous rates vs. average rates. Then, after that I wanted to pull in the idea of limits, which the kids had already seen a little bit of in the context of geometric series. So, I gave the kids this worksheet, and as they worked through it, we set up a huge grid of comparison charts on the board to go over when a "forbidden x value" inside a rational function will be a vertical asymptote, versus a "hole".... and to highlight the idea that a limit is what happens to the theoretical output as you approach those "impossible" x values. We are not done with the worksheet yet, but my hope is that by the start of the next class, the kids will fully grasp the idea that a function with a "hole" somewhere (as opposed to a vertical asymptote) can still have a limit at that breaking point, and many functions also have limits at extreme values of x. That (along with the previous instantaneous rate intro) will prime us for going into talking about the mechanics of differentiation in this next intro to differentiation lesson. One of the things I want to immediately tie into differentiation is that you can check your sensibility of your answers using a graph. I will also right away tie differentiation to the algebraic meaning of turning points. This way, they are immediately exposed to the key concepts in an integrated manner before we do any further practice.

Thoughts?? It's my first time teaching Calculus, so I'm still muddling my way through it while doing my best to sequence the concepts clearly. Your feedback is welcome!

Friday, November 25, 2011

IB: Flash Cards for Concept Retrieval

I admit, I've spent a fair amount of time thinking about what I can do to help my 11th and 12th graders excel on the IB exam. Maybe that makes me a bad math teacher to be thinking about the test so much, but I think my instruction can still be independent of the test, but I have to keep the test in mind as our collective end goal, in order not to veer too far in my attempt to bring good mathematics into the curriculum.

One of my latest conclusions is that doing well on a math test requires two skills: 1. that you can read a problem and retrieve the correct key concepts and/or procedures. 2. that you can apply the procedure without mechanical (algebraic) issues.

(Of course, being a good math student requires much more than that. I want my kids to know where procedures and formulas come from, so that when they apply a formula, it's not just blind application and they don't falter just because one small thing has been changed. I also want them to see the layering of the building blocks and be able to guess at the next logical step. I also want them to see where math is applied, to understand how to model real-world situations using math, and to ask insightful questions. But, ultimately, when they sit down in front of that big math test, I still want them to kick ass and thereby gain the confidence they need in order to do all those other things!!)

To help them with skill #2 (applying the procedure), I've decided that spiraling is very important. Every test I give in class is made up of old IB test problems, and include not just the current topic but also old topics we have already studied. The more we hold kids accountable for skills that we have already learned, the more they will retain them over time. To help them with skill #1 (recognition of cues and retrieval of relevant concept), I've recently introduced to my students the idea of making flashcards for math. It's very simple:

On the front of the card, write down some clues in the problem. (Such as, "This quadratic equation has only one distinct root.") On the back of the card, write down the concept or key math skill that the problem is looking for you to apply. (Such as the discriminant.)

As the kids work on practice problems, they should learn to recognize those major clues in the problems and make flashcards. Each time they sit down to do practice problems, they should first quickly run through their flashcards to re-inforce the cue-concept connections, and then try to approach all problems without looking back at the flashcards. They can also make a few clear example problems on flashcards. (Question on one side, work and answer on the other.) This way:

1. They can study quickly without re-doing the same problems a gazillion times.
2. It makes explicit the cue-procedure connections that "strong math students" implicitly rely on.
3. Over time, if they keep reviewing the flashcards, they will eventually not need them anymore and will hopefully internalize those connections.
4. Particularly anxious or slow test-takers will benefit from having clearly retrievable facts stored on their minds during test time.
5. It's a great way for them to accumulate invaluable study material/improve their study habits over time!

What do you think? Have you tried something like this? I am recommending it to my strugglers in lower grades as well, as a way to reinforce their test-preparation skills. I am helping my entire Grade 7 class build these flashcards this semester, as we review for the December "big exam." I'll survey them afterwards to see how many of them actually used them before doing the practice problems (like I recommended), and how many of them think it actually helped.

PS. Happy Thanksgiving!

Thursday, November 17, 2011

(Semi-Recent) Trip to Poland

I have been so bad about doing updates about our journeying! Part of it is because on our last trip, we didn't really take any pictures. Geoff had imagined that our camera would charge itself via the USB connector, and when that turned out to be not true, we didn't really want to fork out 50 Euros to buy a charger in Krakow. So, instead, Geoff took some pictures on his dinosauric iPhone 3GS, which made everything seem gray and romantic and more or less unrecognizable afterwards. :) (We even tried to take photos in the dark while "catching" someone else's flash. Sad times.)

Anyway, this particular trip happened in early October, during our school's fall break. We were too last-minute to book good airfare deals for going away, so we decided to rent a car to drive to Poland. FYI: Renting a car in Germany as a foreigner is not the easiest. Your non-European license is only valid within the first 6 months of moving here, and even then you have to rent from big companies that carry the international insurance -- which naturally means that you pay a bit more. Also FYI: Parking in Krakow is expensive and inconvenient. Supposedly it's not safe to just leave your German-plated cars on the street, because Polish people like to break in to German cars. So, if ever we intend on repeating this trip, we will be taking a local German train to hop over the border to Poland, and then we will take another local Polish train to Krakow.

Anyway, back to Krakow. You know, Krakow is poetic and beautiful, full of sad history. I didn't really know much about Poland before this trip, but I was intrigued to learn that it's really a country that has been continuously dominated by other empires and countries over the years, from Russian Empire to Prussian Empire to dominance by Austria, followed by a brief period of independence before it got dominated again by Nazi Germany. Also, we don't really think about this, but the Polish people were the second biggest group of victims, I think, in the Holocaust. Auschwitz was built originally to house the political prisoners from Poland, whom the Germans wanted to keep quiet and away from sight. Lots of them were murdered after swift "trials" at Auschwitz. The Polish people were not encouraged to attend schools during the war (and various strategies were used to prevent the intellectuals from gathering and studying), because Nazi Germany wanted to maintain a stronghold over the land and its people.

Anyway, when we went to Krakow, we visited Auschwitz and it was a very intense experience. The tour guides were superb and took their jobs very seriously, and it was very informative but obviously also very sad. When the Nazis evacuated the camp at the end of the war, they destroyed most of the evidence that it had ever been used as an extermination camp. There is only one small gas chamber that still remains, and we were able to go inside. It was very intense. The electrical barbed wire fences are still up and you can walk around and see the gutters that surrounded the camp and see the entrance to the camp and the train tracks that lead up to the entrance to drop prisoners off for sorting. Some of the sleeping quarters can still be seen, and even when we went in October, it was cold. One can only imagine what it was like in the dead of winter, when you are wearing next to nothing. Moreover, there are displays of all of the personal belongings, shoes, and even hair of the victims. The hair was haunting, because they weaved them into everyday products like rugs and resold them. It was deeply sad, and even though there are people who would argue that that place should not allow so many tourists to go per day, I cannot help but think that it is absolutely necessary for them to educate as many people as possible about the horrors that occurred there.

Besides Auschwitz, we also went by foot on our own and visited the old Jewish ghetto in Krakow, like featured in Schindler's movie. Much of the area still teems with old buildings, like they could have been around during the war. We visited an amazing and extensive museum on Poland's role and perspective in WWII, located inside Schindler's old factory. We also went into the small pharmacy that is famous for its Polish owner who refused to move out of the Jewish ghetto during the war, and who helped to hide Jews and to sneak in free medicine for people inside the ghetto.

We also saw the medieval Center of the city. It was beautiful, and filled with cathedrals, happening bars, and yummy restaurants. Our favorite was Marmolada, which is one of a mini chain of 4 restaurants that are all supposed to be different and delicious. This one was fantastic, from the wine to the food to the service to the price. It's also a 1-minute walk from the city center, right off of one of the little alleys.

Lastly, we visited a cheesy tourist trap of a salt mine in Krakow. You'd never think it, but there is a beautiful chapel that is carved entirely out of salt in the depths of a salt mine, made by 3 professional miners who did this as a hobby over the course of seventy years. They made it complete with statues and chandeliers and -- you would never believe this -- a breath-taking statue of The Last Supper, carved into the walls. Talk about a personal hobby project!

Last note is that the Polish roads are supposed to be horrific -- even my coworker's wife, who is herself Polish and native from Warsaw, said so. The funny thing is that on our way to Poland, while we thought we were still on the German side of the border, the highway started to get really bumpy. You couldn't really see any difference, except the car was going up and down and up and down as though the road was filled with potholes. We were joking that perhaps we were already in Poland, and sure enough, in two minutes we started to see road signs marked in Polish letters...

The next big thing I'm looking forward to: German CHRISTMAS MARKETS!!!!

Scatterplots and Patterns

I had a pretty great experience with my Grade 12's today using these made-up data sets to discuss with kids how to find multiple modeling functions and then how to narrow them down to one using either asymptotic behavior or the meaning of that type of function. I used it in conjunction with the function types handout that I had given them before, and they were able to ask me some good questions about the differences between functions and how do you know which one is the most appropriate, when a few look very similar?

Even though I wish that they could have come up with the handout information themselves about various function types, it was very valuable to hear them ask those relevant questions, and some of them (the ones who had brought laptops from home) were able to play around with plotting points, setting up sliders for parameter values, and then estimating best fit equations all in GeoGebra. Pretty great, for one day! (The rest of the kids plotted points in their graphing calcs and practiced looking through the functions to find possibilities.)

In the end, I passed out our official IB task and they looked over it and nobody panicked. I think that is a good sign.

Wednesday, November 16, 2011

An Unexpected Math Encounter

I went to a cabaret / dance thing last weekend. (It was more of a dance thing than a cabaret thing, which was totally cool except that I had thought it'd be more of a cabaret thing and wore heeled boots and danced with my boots on a slippery wooden floor all night.) At the entrance, there was a sign:

"Pay 9,50 Euros OR toss a die to try your luck. If you roll 1, you pay 7 Euros to get in. If you roll 2, you pay 8 Euros to get in. If you roll 3, you pay 9 Euros... etc. Up to a possible 12 Euros for entrance."

That made the math teacher in me happy to know that somebody paid attention in their math class. :)

Tuesday, November 15, 2011

Fun with Ratios and Thinking About Functions

My middle-schoolers are coming along nicely in their conceptual understanding. I would like to share one project that my 7th-graders are working on currently. It's an old ratios assignment from when I taught middle-schoolers way back when, but still one that gets kids all riled up and excited, apparently.

First, I want to share a neat little trick with teaching ratios. My first and second years of teaching, I had a lot of trouble getting kids to "see" how to convert part-to-part ratios (such as girls:boys = 3:2) to part-to-whole ratios (such as girls:total = 3:5). I used all these hands-on manipulatives and guided exercises, but to no avail! My third year, I finally figured it out. All this time I had been trying to explain ratios, when instead I should have been letting kids observe ratios, because ratios are an intuitive concept!

So, nowadays I teach ratios problems using tables of values. The tables always start with zeros, because I want to continuously reinforce the connection between direct variations and ratios problems.

I've noticed that this way, no matter if the kids are starting with a part-to-part or part-to-whole ratio as given in the problem, they can always manage to find the other missing parts. And also, they can see how every part (or column) scales equally, thereby leading naturally to the idea of a scale factor. When the problem then asks, "How many boys would there be, if there are 966 girls?" they can figure out easily which column 966 goes into, and then extend (fairly easily, with a bit of practice) the idea of scale factors to find the other column values. --And this is with my Grade 7s, who had never before seen any ratios problems! In the past, when I used the same method with my 8th-graders, I didn't have to teach them anything at all; they could just observe the pattern and figure out the scale factor shortcut when they felt tired of extending the table tediously.

Anyway, following some practice of the table method, I finally introduced cross multiplication. The kids were mad at me; they thought that setting up and solving proportions was way more difficult than making the tables! I made them practice both methods so that in the future, should a teacher require cross multiplication and/or the problem involves decimals, it'd still be in their arsenal. But, I am happy that their conceptual understanding is strong enough to want to replace the cross multiplication.

In any case, back to the ratios project. The kids are given the specs of an original shape and partial specs of a new shape. They need to use their understanding of part-to-part and part-to-whole ratios in order to find the scale factor and to scale all sides appropriately. In the end, they measure and cut out both versions (initial and final), and in groups of 3 write out an explanation of the entire process -- which everyone has to agree upon -- and create an explanatory poster.

I like this project, because it allows me to differentiate easily. Some groups have scale factors that are whole numbers, and others have scale factors that are unit fractions (ie. 1/n where n is an integer), while others have scale factors that are non-unit fractions. Since they're not told what the scale factors are, finding it can be a bit tricky when it is a non-unit fraction.

Anyway, so far, it's going swimmingly! The kids have almost all finished measuring and cutting out their two shapes and are working on the final written explanations. If that sounds like fun to you, here are the project prompts I used this year (excluding the instructions for the posters) and the warmup exercise I used to introduce the idea of geometric scaling.

Love projects!

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For Grade 8, we are wrapping up two application assignments of the functions we have been learning. I did Dan's cup-stacking activity with them and they enjoyed it immensely -- 3 or so groups came within 1 cup of the final result! I also did a traditional profit and revenue assignment with them. It was the first time I felt like we were getting into complicated application, and although it surely challenged them conceptually, I could see that some of them, at least, enjoyed the discussions about why it makes sense that profit and revenue are both shaped like a parabola.

Today, I taught them to graph linear functions. But I did so by tying it firmly into the meaning of slope and y-intercept inside a word problem, so that every time they look at an equation like y=3x/5 + 2 they will think to themselves that 2 is the starting value (such as # vacation days at the beginning of the year) and 3/5 is a rate -- 5 describing "how often" it happens and 3 describing "what actually happens." (For example, every 5 months you gain 3 vacation days.) It was the first time I did graphing by connecting it to my Mad Libbs worksheet on interpreting linear functions, and it worked really beautifully!!! Not a single kid was lost in the graphing of equations, when they were just thinking about the meanings of the rate and the starting value and putting that into graphical form. Kids were even figuring out for themselves the idea of rise and run in a fractional slope, even though they had never learned this before!!

:) Yay to experimenting. Small changes, big impact!

Addendum 11/16/11: Today I gave my kids a bunch of graphs for them to write equations for. Using just their understanding of the meanings of parameters inside the slope-intercept equation as related to word problems, they were able to easily look at graphs and to write those equations without messing up! This has been the smoothest teaching of writing equations EVER.

Monday, November 7, 2011

On Being Truthful

Today, when I was grabbing something out of my closet, my 7th-graders came into my room to get ready to start our class. One of the last kids to come in did not see me behind the open closet door, and she cheered loudly when she thought that I was absent. She was immediately embarrassed when she realized that I was present (and standing quite close to her), and her friends all laughed.

Innocent mistake. I would have just let it go, except when class started a few moments later and a few kids were still giggling about it, I steadied the class and I asked her if she had something to tell me about how she felt about my class. She said no, that [they] like [me]. And I added softly, "Because I would never cheer even when a kid who's really difficult misses my class. What you said... hurt my feelings."

I wasn't mean about it, even though the kid had embarrassed me (intentionally or not). I think that's something that I've learned to do with grace in the last few years. During the class, the kid tried to sneak me a couple of quick apologies, and then after class she hung around and explained that she was sorry that she had been offensive, that she hadn't meant for it to be hurtful but that she was hoping for some free time today to work on something during class.

I appreciated her for her apology. As a teacher of middle-schoolers, I understand that I cannot take these things personally. But, I think it was the first time I embraced the truth and frankly told a kid that what they said was hurtful to me. And hopefully, the next time it will make her think twice about being inadvertently hurtful to someone else.

I am trying to model for my kids the kind of adult I hope they would grow up to be. That's the most difficult task of all, because I'm not sure I am that person all the time. I think this year I am closer than the years before, because I am much more aware of it. But I've still got ways to go.

Intro to IB Type 2 Task

This entry is for you IB math(s) teachers out there. I pulled together an introduction slideshow for the Type II task to make it super clear for kids what the mathematical modeling process needs to look like.

Here is the Prezi, and if you wish to make it fullscreen, just click on the "More" button in the lower right corner of the Prezi presentation to see that option. The plan is to walk the kids through the various steps of modeling, and then to jump right into helping them install Autograph (30-day trial version) on their laptops, and then to play around with the program a bit in class, and then to go over the function basics again, and then to take them through an easy sample task.

This is, of course, meant to be used as a follow-up discussion assuming that your kids have already seen these awesome demo videos Part 1 and Part 2 that I had found on the web.

PS. Also, to help kids keep track of all the function types and features, I made this rather comprehensive handout that you are welcome to take. I know, it's not perfect. I wish kids could just all look at data sets and figure out on their own what functions make sense based on their own awesome mathematical understanding, but I don't think I have enough time with my 12th-graders to make that happen overnight, so I'm giving them this as a bit of help, and I will go over it with them as a class to make sure that they are taking notes and trying to understand the important features of each graph. Better than them going on the internet and copying and pasting things they don't understand, no??

Wednesday, November 2, 2011

Responding to Student Needs

I've been slowly reading Lost at School as part of a professional book club at work. Honestly, I don't think much of the book is ground-breaking stuff, but it's a nice common-sense teaching book to spark some common-sense teaching discussions. It provides a good focal point at work for discussions about what is important to all of us, rather than discussions about our individual concerns.

Here is a quote that I liked that helps me view my current experience through a different lens:
Good teaching means being responsive to the hand you've been dealt.

It goes without saying that each group of kids is different. The task with each group is to get a handle on its collective strengths and limitations and work toward building a community where each member feels safe, respected, and valued. But that takes time and concentrated effort. It doesn't happen by itself. And it looks different every year. That's what it means to be responsive.

It also goes without saying that every individual in a classroom is different. [...] The ultimate challenge is to be responsive in both ways -- to the group and to the individuals in it -- simultaneously.

I think this really nicely outlines all of the things that are on my mind constantly, as I struggle to grow into being a "good" teacher for my kids.

* My Grade 7's have a few lagging performers who haven't yet figured out that math is important and they need to come see me outside of class for help, so (in response) I weave all of their review of past topics that they're still weak with into our normal class. Every test is cumulative and hits every past topic, and we review accordingly beforehand. I also send daily emails home to let parents know when their kids miss an assignment. And, once every few weeks we play a game that reviews an old skill that I want them to fully master.

* My Grade 8's are a bimodal group, so I need to balance between keeping the really advanced kids challenged and giving the strugglers time to work on their factoring skills when the leading coefficient isn't 1. I do so by introducing every factoring tool possible (looking at graphs & using quadratic formula), and also giving the top-top kids extra graphing calculator assignments to work on independently.

* My Grade 9's are the lowest group in the grade, so there are kids in the group with serious language issues, others with no mathematical memory past the current day, and a few kids who are working very hard to bridge the gap of their learning. I need to provide tasks that are accessible to all kids and allow them to work at their individual paces, and I give them free reign to correcting/re-doing all old assignments as many times as they want. Frequent assessment with clear skills expectations is key to making sure all kids are given regular feedback, and I've met half of their parents already to discuss my concerns about the kids.

* My Grade 11's are a mixture of returning kids and new kids to the school. The returning kids are much more experienced with the first topics of the year, but to ensure that all kids have a fair shot at the IB exam, I'm teaching them all from scratch to make sure the new kids get proper training. Again, it's a balance act of approaching the topic from many angles to ensure that the returning kids are challenged and enriched and pushing the new kids along with some urgency to make sure they do a bit of extra homework to keep moving at the same pace as the others. In the longer run, I've worked out with a colleague that she can transfer the kids who need a bit more nurturing into my group, in exchange for moving kids who wouldn't mind/could handle moving at a much faster pace into hers. We're both happy with that arrangement and think it will maximize the benefit to all kids.

* My Grade 12's have a lot of gaps in their knowledge. They basically don't know/remember anything they are supposed to know from Year 11, and so I've been interleaving as much of old material as possible into current topics to help them review. I know that in the spring, I will have to do some very heavy-duty concept-mapping and explicit learning strategies like algorithm flashcards to get them familiar with the basic concepts of each topic, before we can start doing integrated review and test prep. I am prepared to make that happen and I have a plan for how to help them be successful. (It'll involve topic-based worksheets / individual tests when they're ready / moving on to individually review the next topic when both they and I agree they've mastered the basic ones, or repeating the cycle until they do improve.)

It's stretching me professionally to consider the various academic and emotional needs of my many groups and to attempt to address individual needs within each group. But, I am loving the challenge! :)

(...Now, if only someone could tell me if it's having any actual effect on the kids...)