Saturday, March 31, 2012

Thoughts About Cultural Sensitivity

A recent incident has stirred up some feelings for me regarding cultural sensitivity in schools. As a teacher, generally I feel responsible for the character education of all of my students, so whenever an occasion arises -- in class or otherwise -- where I see that someone is being offensive or hurtful to others (for whatever reason), I address it firmly and make sure I send out a message that that type of behavior is absolutely unacceptable.

As a person of proud Asian descent, I am particularly bothered on a personal level when I see actions around the school setting, that stereotype and/or demean Asian people. But, countering this is not always so easy, especially because in many cultures besides in America (surely in German as well as Salvadoran cultures), there is much less cultural sensitivity in general. I have heard a colleague say things to me that would be considered highly offensive if said in the States, even in private, much less in a professional environment. As a teacher in El Salvador, you live with the reality that one boy out of at least each class is nicknamed "(el) Chino" because they have slanted or small eyes. One "Chino" from my Precalculus class was a blond boy; the nickname was not meant to be hurtful or disparaging. Other culturally "acceptable" nicknames are "(el) Gordo" (the fat one), "(el) Chele" (the pale one), etc. Totally unacceptable in the States but totally common-place in Latin America.

So, the line becomes gray in other cultures. --Or does it?

That is what I have been thinking about the last few days, since I had to deal with a kid who repeatedly put offensive images about Chinese people on a slideshow that he was trying to put together about China. As it turns out, neither the boy's mom nor the other colleague involved in the incident thought the pictures were so bad. (The images made fun of how Chinese people eat cats and dogs, and also had bad English phrases on them like "Sum Ting Wong?") The pictures were so offensive to me that I had to delete them off of my desktop immediately after sending them along to another adult, so that I could get rid of the negative feeling that the images caused me. When my colleague saw the pictures, their first reaction -- in my presence -- was to laugh. Is it my job to educate them? Where does my role as an outsider fit in, in terms of pushing back on these cultural sensitivity issues?

On a semi-related note, recently, an acquaintance mentioned on Facebook that she received a letter from her bank with a Chinese card tucked inside. She was offended, because she's actually Vietnamese. From an outsider's perspective, maybe you would think she overreacted, but unless you are from a country that constantly gets lumped into another country's ethnic group, you cannot begin to understand how she feels. (Her last name is Nguyen, by the way. That is the most common Vietnamese last name. The person at her bank who made this Asian publicity stunt needs to feel bad; that is like addressing a Kenny or O'Connor and saying they are Swiss.)

So, this is what I know: Racism is real, and it is hurtful. Whether or not a person is actually racist, their behavior and choices speak volumes for them. If we do not educate our children about what is acceptable and culturally sensitive, then they will grow up to be adults who help to propogate harmful racial stereotypes.

So, again, my question is: What is my role as an educator to help to stop this? Does my role change in an international setting? If you have thoughts, please feel free to add them. Otherwise, this is something I guess I'll have to figure out for myself, because it is important and worth thinking about.

Thursday, March 29, 2012

An Epic Week

This week has been epic. It was a week when I tried to do something for everyone. It was my last week with my Grade 12 IB students (since the next two weeks will be Spring Break, and after that I'll be absent from school for a week to chaperone a trip to China, and after that they go on a week of "study leave" before the IB exams); my last week teaching material to Grade 11 students before their semester exams; and the week that my MS principal and I looked through and picked two Grade 7 buttons business plans to invest money into. All of my 7th-, 8th- and 9th- graders had to come to a conceptual stopping point before vacation AND they had to be sufficiently solid in the new concepts that they could do independent practice/review without me, during my week-long absence after the break. It was a week when I promised to draft up 3 different semester exams so that I could help my colleagues agree on final exam content and dates for three different grades, so that I could send my own students off on their Spring Break with all of the necessary studying information. This week, as a Grade 12 team, we had to send in audited samples of graded student portfolios to the IB Organization, and internally, we had to submit estimated IB grades for the 12th-graders. This week, I finished babysitting/monitoring the enrichment research projects for the kids who are going to China with us, and they successfully presented their projects to their parents at the last trip meeting during Monday evening.

In the end, the week was smooth, without a hitch. It has wound down beautifully, and even though I am running on 5 hours of sleep today (went dancing yesterday), I feel very satisfied with what I was able to accomplish during this week.

And, oh boy, am I ready for Spring Break! I am looking forward to being in the States for a week, followed by being in Beijing and Shanghai (while chaperoning students) for 10 days and SEEING MY PARENTS!!! and eating some yummy soup dumplings. And when I come back, it'll only be a short sprint (punctured by another whole-school field-trip week and various holidays) until the end of the school year!!

Monday, March 26, 2012

Functions Fun with 9th-Graders

I have to say that as a classroom orchestrator, my weakness is in setting up DRAMA in my lessons. I do, however, put on a great big show every year if there is a group of kids to whom I have to introduce functions for the first time. I do a big show of function machines (not a unique idea, obviously), by gradually feeding input numbers written on post-its into a "Function Machine" (this year, it was a black plastic bag with a "Function Machine" label on it) and flipping the post-it over inside so that the kids can be delighted by the output value that comes out. You might think the kids are too old for this, but NO! This year, one of my Grade 9 boys had to stifle a giggle when the first number transformed inside the function machine. His eyes got so big. Together, as a class, we fill out a function diagram on the board ONE post-it at a time (and the kids copy it down in their notes), so that they can see how the function diagram visually represents the mapping of the elements. At some point they start to feel very antsy to make predictions for the next outputs once they start to see a pattern between input-output pairs, so that sense of anticipation really teaches them about the predictability nature of functions. At some point, when they absolutely cannot stand how smart they are anymore, I ask, "So, if we put in some number x as the input, what will be returned?" And when they give me the answer, I both write x ----> x + 2 (for example) as a pair inside the function diagram, and underneath I write in a parallel formal notation:

f(x) = x + 2

And at that moment, with a tiny bit of clarification, it becomes crystal clear to them what each part of the function notation represents. (Obviously, it still takes practice to gain confidence/familiarity with the notation, but I think putting it directly underneath the x ----> x + 2 notation definitely helps to clarify why there is an X on the left side of the equal sign.)

Then, we repeat several other dramatic sequences (with increasingly complex patterns), each time referring firmly back to the definition on the board of a function HELPING US MAKE PREDICTIONS; each X value has exactly 1 Y value. I try to vary up the representations for the later examples, using tables or ordered pairs, so they can realize that all representations are actually equivalent. (Towards the end, to help them focus their understanding, obviously we look at cases of both shared outputs AND two outputs for same input, and we use our function definition of PREDICTABILITY to try to evaluate which case is a function and which is not.)

There are other things I do on Day 1 of functions that I think also help to clarify the definition of a function (such as a sorting activity involving cards that have maybe-functions printed on them), but the DRAMA! is what is so fun about this particular lesson introduction. I absolutely love it every year, even though it's so simple in nature and in foresight always seems potentially very cheesy. --And, trust me, I know that if my 9th-graders are glued to my every move at the board, it's a lesson to keep for the long haul. :)

Do you have your own favorite dramatic introductions of new concepts??

Sunday, March 25, 2012

Cool "New" Blogs

Maybe this is news to you, or maybe it is not, but Fawn Nguyen has a really nice blog about math teaching and miscellanies.

I am kind of a blog junkie, and have always been. To me, what type of content makes a teaching blog nice? --Specific discussion about activities, and why they structure them as such. Honestly, I am too concrete a thinker (and am generally too busy) to read abstract, philosophical discussions that do not give specific illustrations. Personally, I also appreciate it when bloggers don't complain continuously about the same thing, because I believe that we all have our reasons to be negative, so there is no reason to dump our own negativity onto others. I think it's a nice rule of thumb to say that if you point out that something doesn't work or isn't ideal (which is fair to do from time to time), that you must offer a concrete alternative that could be feasible and more effective. Our school admin mentioned this in the beginning of the year as a general courtesy to keep at staff meetings; I try to keep that courtesy online as well. On the receiving side, I definitely recognize teaching blogs that leave me with a heavy feeling, versus teaching blogs that make me feel inspired and happy.

Fawn's blog is specific, positive, and it also shares some personal tidbits, which is also nice. I personally enjoy it when someone's blog goes a little bit beyond the teaching and shares a little about their own lives, because if I care about their teaching, inevitably I start caring about their lives as well...

Another blog that I also think is very nice (but that has been around now for a little while now, actually) is Amy Gruen's Square Root of Negative One Teach Math. Amy's blog is not pretentious and is altogether lovely, and her love for her students shines through effortlessly.

Also, the awesome Tina Cardone is as fun online as she is in person!

Enjoy! If your affinity for blogs matches mine (in terms of what type of content you prefer), I would love it if you could comment and leave me with a few suggestions of new blogs to follow.

Saturday, March 24, 2012

The Other Skills People Don't Mention

Valuable lessons my mom taught me when I was younger:
* Sometimes she would call me from work, leave me a string of numbers (this was before the days of email or texting), and before she hung up, if I had asked to repeat all the digits back to her, my mom would be exaggeratedly pleased. She said this showed a care of mind in completing tasks, and therefore a general competence in doing any job. In various situations similar to this, she taught me to always tend to the details.

* One summer when I was 16, I had a part-time job at a fast-food restaurant and I had to frequently clean well past my shift without pay, because the (cleanliness) expectations were high and our store manager tried to squeeze work out of us without wishing to pay anything extra. (And my work ethic was too good to just leave after I clocked out, if the job wasn't finished.) I was very upset about the store manager systematically taking advantage of us, and my mom told me that I needed to learn to quit jobs on principle. She said that if I couldn't quit then, I'd never be able to quit jobs on principle when I had a family to feed and bills to pay.

At some point in my first internship, I realized that:
* People recognize and appreciate motivation. Even though I was only a QA intern, I wanted to program. I always stayed an extra two hours at work that summer, to do my own project after work. On my break, I would chat about the project with other colleagues / adults, and they were always impressed by my motivation. (At the end of summer, they all wrote me glowing recommendations.)

At some point in my first job (as a software engineer), I realized that:
* Your manager is responsible for your happiness. If you are unhappy, they are liable. So, your job as an employee is not simply to sit and endure all the things that are thrown at you; if you need something (additional resources, time, support, etc), you need to feel brave enough to say it and to expect that something is done to address it.

* Soft skills like the ability to work with people and the ability to ramp up on a multi-faceted project or situation quickly, are just as important as hard skills you can write down on a resume, even in a technical situation. I was never the most technical person on my team, but my supervisors always thought I was very valuable.

When I became a teacher, one of the first things I realized was...
* People will bully you on the job. A school is like a self-contained sphere, wherein you need to be a strong enough person to stand up for yourself. If you do it once, and then you do it twice, and you never let someone walk over you, then eventually that person will stop trying. But if you don't do that, it will get worse.

These are still valuable work skills that I carry with me. As a teacher, often I find myself needing to "manage" horizontally (to get my peers to work together), to "manage" upwards (by stating my concerns or by gently pushing back), to manage logistics, or to understand that I always have a choice to stay at or to leave a job (which, again, only matters to people if you are motivated and it is evident that you try to do a good job).

Sometimes I think how I work with adults is just as important as what I do inside the classroom with children, because if I don't do those things well outside of the classroom, then my work with kids is made infinitely more difficult. Do you teachers agree?

Thursday, March 22, 2012

What do we NEED to know?

I am very much enjoying the itty-bitty geometry in Grade 7! We have been working a lot with hands-on investigations to develop concepts around triangular areas, shearing, and square roots.

For tomorrow, since it's the end of a week, I am going to use a short discussion to re-focus their attention on the idea "What do we know? What do we NEED to know?" which I think is so key to understanding Geometry.

So, I made this powerpoint file to guide that discussion. (You have to actually play the slideshow in order to see the animations properly.) We're going to look at different basic shapes, initially without any values labeled; I'll have volunteers go up to the board to show what values we NEED to know in order to find the areas, and then I'll click the mouse to reveal a bunch of values (including extraneous info), to actually calculate the areas. The last slide will be an introduction into quadrilateral areas, which we will explore next via hands-on cutting and pasting.

(I am very happy that all of my 7th-graders are drawing rectangles around triangles to find areas, and that they are un-doing shearing in order to find areas of oblique triangles! Way to not memorize formulas!!)

Any last-minute recommendations??

Wednesday, March 21, 2012

Drilling Exponents

After our successful intro to exponents in Grade 8, I feel that the students can mostly articulate how to simplify exponents and why, and that they were ready for some more focused drill / practice. We are about halfway through the exponents mini-unit -- haven't yet introduced negative or fractional exponents yet, but they should be pretty OK with everything else by now.

Since I have been looking for creative ways to stay within my printing quota, I thought of a move-around activity for my students. So, I put up a bunch of different questions around the room (a total of 22 problems) on index cards, and asked them to move around to try simplifying each one. The answer is on the back (written upside-down so that when they flip it over along the top edge, it becomes right-side up), so they can quickly check their own answers to know if they are on track. Their task was to do as many as they could, to mark the ones that they had gotten incorrect*, and to ask me for help if they really cannot figure out, even after looking at an answer, how that answer was obtained.

*I had asked them to mark the ones that they initially got incorrect, so that in case they come to me asking for more practice problems later, I would know which type of questions to make for them to address their individual issues.

It was great! Kids really got to move at their own paces, and most of them finished all 22 questions and started a new assignment. Also, because they were already moving around the room, kids who don't typically collaborate during class started working together in random corners of the room. I think the on-the-spot answer-checking was also good for the weaker students, because they can address their own gaps (and also move at their own pace) without feeling insecure about it, and if they did get something correct on the first try, it was a nice boost for their confidence.

Exponents, unfortunately, is such a hellishly boring topic. The move-around activity and also speed games** help to make it a bit more exciting, I guess. (Because we haven't played too many games this year in Grade 8, the idea of group games is still fantastically fun for them...)

**I think I wrote about my typical games format a while ago. Very simple: Two teams, each team sends up 2 people at a time. The teammates can collaborate at the board but when they raise their hands with an answer, only one will be chosen "at random" to explain the answer. Most of the time the teams figure out to pair a strong student with a weak student, and they figure out also that the weaker student will probably be asked to explain, so at the board the stronger student is trying to explain the concept to the weaker teammate -- perfect peer learning opportunity! It also keeps rotations faster so there is less crowd idleness. Problems that are unanswered at the board go to the crowd for 0.5 point, so that is an additional incentive for the audience to stay looped in and to try the problems at their desks.

Saturday, March 17, 2012

My So-Called Remedial Class

I've been teaching a "remedial" Grade 9 class this year. It is actually my first time teaching an actual remedial class! In all the other years of teaching, either I taught heterogeneous classes (sometimes with very weak students mixed in), or I taught classes that were meant to be mainstream or even honors classes within a streamed setting.

I am teaching a "remedial" Grade 9 class. In the beginning, it was very rocky. In between me being a new teacher at the school and the kids being sorted by levels for the first time in their lives, there was a lot of upheaval and a lot of unhappy kids and parents. In the end, I ended up with 10 students (in my smallish classroom). They were very weak and all the boys could not sit still and they refused to speak English in the classroom. (I've dealt with very severe ADHD at my last school, with kids falling out of their seats and 5 ADHD kids in a class of 20, so hyperactivity is not really a problem for me.) And they were rude to me out of anger about the situation, so they made snide German comments to each other daily. They wanted an expressway to a "normal" class and they did not believe me when I said that I would be teaching them normal grade 9 material, but in a different way that allows them to cover more ground in a year in order to catch up. There was a lot of hurt and a lot of anger/resentment, so the class was an uphill battle from the start.

Well, fast forward to now. It's March. I just gave a pretty hard trigonometry test to these same "remedial" class kids. On this test, almost every problem was a word problem, for which they would have to draw their own diagrams before even getting to the solving part. It involved a lot of thinking. --AND THEY DID GREAT!!! Generally speaking, they have been doing good work for me this year, and it's more and more apparent to me that the boys in the class simply needed a different way of teaching that fit their active personalities. Maybe what they had needed was a teacher from the Bronx who would give things to them straight. But, whatever it is, these kids are WORKING, and even a boy who had never bothered to do any math in his entire life up until January, has been trying very hard and got a majority of the questions on the previous trig quiz, correct. Even though they were all multi-stepped!! It took him twice as long as everyone else to do that, obviously, because I think it was the first time in his life that he actually sat down to try to gather his thoughts and to REALLY try every problem on an exam after having worked in class for 2 weeks straight.

I feel so positive about this. One of the boys had gotten perfect scores on two consecutive quizzes, so now there is even a level of healthy competition going on amongst friends.

I am still trying to figure out what the deal is with my girls, however. My boys, I think I've got a handle on. My girls who work consistently, on the other hand, are not always able to show that off on the test. (I was able to promote two girls to the mainstream class at the semester, so obviously this statement does not pertain to all my girls. But, I got another girl in exchange who was moved down from the mainstream class...) So, the work is still cut out for me, but I most definitely look forward to this class more and more, and I am excited that I will get to keep teaching them through next year, now that I have an actual relationship with each of them that I am building on top of. (They have since stopped being rude to me. Well, most of the time anyway. They don't call out, they raise their hands to explain stuff, they listen to each other most of the time... Many of them even do weekly homework, 20 problems of their choice from the textbook!!)

PS. You can see a copy of my said trig test here. Sorry but the font is small -- for two reasons: one is that I am almost out of printing quota for the month. The other is that I created this on a Mac instead of my normal PC, and so I had to send it to myself as a PDF and couldn't modify it afterwards. Doh.

But see? If you look at this test, I don't think you (nor I) would expect half of a "remedial" Grade 9 class to have aced this. It is true: 3 out of 9 kids missed nearly nothing (only rounding errors and such), and 2 more made only very minor calculator-entry errors. That is not bad!

Life in Germany

I haven't updated much about life in Germany in a little bit, because honestly, I've been mired in work trying to get through the first season of preparing my students for the IBs. (They leave school at the end of April on a week long "study leave" before the exams begin, so I'm certainly not out of the thick of it yet.)

But, little updates about Germany. Nothing big because we haven't traveled much recently, but just little things:

  • Did you know that Germans typically get married while they're having the public ceremony at the city hall? As in, instead of that part just being a paperwork thing, the entire family goes down with them to witness that moment when they get married. In fact, the entire wedding is planned around that appointment at the local town hall. Of course, this creates a bit of issues because Germans really like to get married on "easy to remember" dates, such as 4/8/12. (That means, by the way, August 4. European-style dates, as I now prefer. It really makes more sense if you think about it, than mm/dd/yy.) My friend Mandy, for example, had to reserve a place to hold her wedding reception a year in advance, and then only 6 months in advance of her wedding date could she go down to the town hall to make an appointment for her wedding. The line was very long -- there were people who had arrived at 5am to wait (in sub-freezing temp), even though the town hall doesn't open until 9am. She got the last appointment available for 4/8/12, and there were still about 20 people in line behind her who didn't get their appointments made. I asked her what would have happened if she didn't get that appointment, and she said that basically she would have had to try another town hall, and then maybe another. But, this is not good because that means that they would have had to hire a bus to take all of their family members to possibly a far away town to get married, and then to take them back to the wedding reception area. Phewey! And, by the way, most German weddings are very small and only involve family and maybe a couple of very close friends.

  • In terms of German schooling, I've heard from a math teacher that her child is in a public German school, and they start streaming / grouping by levels in as young as 4th grade. By the 4th grade, kids are already carrying planners around with them all day and there are very high expectations of independence.

  • Of course, this is tied to cultural expectations in general. On the streets of Berlin, I often see very young children (maybe 3 or 4 years old) crying while riding/wobbling on their bicycles for the first time down the street. Their parents are typically at least a block ahead of them and not stopping to wait and nurture the kids as you would expect American parents to do. Of course, for months before that (maybe since they're 2 or so??), those same kids have already been running down the street on their little training bikes going as fast as their parents on the big bikes.

    Training bikes look like this. The kids glide on them so fast even without pedals, they're practically already riding bikes down the street.

    But yes, it's definitely tough love. That's why one of my German colleagues could not understand why a 6th-grader's parent did not want us to just leave the kid at a cafe in a train station by himself to wait for his parents to come with his passport before hopping on the train to catch up with everyone else who would have already been on their way. To Germans, 6th-graders should be perfectly capable of doing these things.

  • And if you have been following the German scandal with their former President, Christian Wulff, you might know that many Germans are very happy to see him step down. The President of Germany, different from their Chancellor (Angela Merkel), is more of a figurehead than anything else. They say that he is "kind of like the Queen of England, except he is elected." Because of that, they expect him to be basically perfect. So, even though his scandal involved possible corruption from when he was a governor, he had to step down. Our German friends were following this bit of news closely in hopes that he would.

  • I recently witnessed the German BVG people (sort of "subway and bus police") come around and check tickets on a bus. It was very funny because the bus was very crowded and the stops were too frequent for them to be able to catch people getting off the bus right away when they had gotten on. They ended up jumping off the bus at the first stop, because they thought someone who didn't have a ticket was running away. The whole thing was pretty comical, and I wish they would just stick to checking tickets on the subways, which seems to work pretty well for them. Busses are just too chaotic for doing that kind of thing. ...But, I have to also say that I think people should just buy bus tickets legitimately. We need to support the public transit!! The German transit system is by far the best I've seen!

  • Something for me to investigate is cigarette laws here. Geoff and I met a Marlboro marketing guy when we were in Turkey, and he said that in Germany it's legally allowed to post cigarette ads everywhere. In fact, Marlboro right now has a huuuuge campaign called "(May)Be" that has the word "Maybe" but the first part crossed out. I never understood what it meant until he explained that it means that instead of being a "Maybe" smoker, you should just "Be" a smoker. How terrible that this is allowed!!

    This definitely has a negative effect on the young smokers. I regularly see teenagers smoking around schools. I have also seen them at the bus stop rolling their own cigarettes, which is interesting because it looks like something else. I can only imagine that if you roll your own cigarettes, the filter doesn't work very well and you're getting even more carcinogens into your body.

    I don't mind when adults smoke (as long as they are not disgusting about it), but I think it is really bad that we let kids, whose bodies are still growing and whose minds are not yet ready to make their own decisions, be exposed so readily to cigarettes. :(

  • On another note, I've been learning German, slowly but surely. I am really glad that I make weekly appointments with my private tutor, because if I didn't, I surely would not feel motivated to be doing stuff every week on top of being incredibly busy at work. But, with her I feel that my reading comprehension is certainly getting better, and my understanding of the German grammatical structure is as well. On top of it, I've now finished the Pimsleur Book 1 audiotapes and have begun Book 2. I am happy because I think I am moving along about as fast as I did when I learned Spanish, and I was certainly comfortably conversational in Spanish by the time I had left El Salvador. Right now I can make broken sentences to say to my German teacher to tell bits and pieces of a story, but sometimes I still lose patience and switch over to English. By the end of the year, I hope to be able to say everything in German!

Friday, March 16, 2012

Pancakes! (and Function Compositions)

I was reviewing composition of functions today with some of my students as part of a bigger review on function basics. The example question I used, a simple one, specified that
f(x) = 2x - 1 and g(x) = sqrt(x) and asked them to write the equations for (f ○ g)(x), (f ○ f)(x), and (g ○ f)(x).

I showed them how to re-write the composition notation like this:

(f ○ g)(x) = f(g(x))

and then I asked, "Is this composition primarily an f function, or primarily a g function? If you could figure that out, then writing the equation will be really easy..."

The kids stared at me in silence.

I tried hinting at it, "Let's think about which function occurs first and which occurs last."

"G occurs first, followed by f taking in the output of g as its input," so said the kids. But they still weren't able to state what the primary function is. The hint was too esoteric.

I tried again with a silly analogy: "OK, let's say you first make someone really fat, and then you smoosh them down and flatten them out like a pancake, then are they primarily fat or primarily a pancake??"

Kids giggled and mumbled more or less in unison, "Primarily a pancake!"

"OK then. If g occurs first and then followed by f, what's your primary function here? Which one is the pancake?"

Kids mumbled, "F is the pancake."

"Are you sure? Which is the primary function here?"

"F," they said. They're still not sure how this helps them write the formula.

So, together we copied down the equation for f, since we know that this is going to be the basic form/framework for our composed function:

f(x) = 2x - 1

But since we know that f is no longer just taking in a simple x value but taking in the entire function output from g, we can replace the bold parts above with g(x) and its associated formula as shown below:

f(g(x)) = 2*sqrt(x) - 1

Easy breezy. (On the board, by the way, I made sure to circle the bold parts and to draw arrows between the equations to indicate which x's got replaced with what.) The whole thing took less than 5 minutes to teach and all the kids said they now understand both how to do it and the concept behind it. (I gave them two more questions to try, just to be sure. They had to do
(f ○ f)(x) and (g ○ f)(x) on their own, and they did them both quickly, correctly, and without any doubts.)

I think review sessions for small groups of struggling students are helping me to become a more efficient explainer altogether. I find myself thinking of different ways of explaining things than how I normally structure it in class.

So yeah, pancakes! They are the magical ingredient to teaching function compositions, apparently.

Thursday, March 15, 2012

My Love Affair with Geoboards

I have to say that I am a BIG fan of Geoboards. It is great fun watching kids do stuff on Geoboards to get a tactile feel for what area and perimeter actually mean inside formulas. I've only been using Geoboards for two years, so I still have a lot to learn, but things I think it can immediately address:

* Many kids don't really understand what "unit" means and why units are different from points. For example, hand a Geoboard to any kid who is already familiar with the definition of area and how to find rectangular area, and ask them to build you a rectangle with a certain "easy" area, say 24 squared units. I guarantee you that MORE than half of the class will immediately build you rectangles with the wrong area, because they are counting pegs (or "points") instead of spaces or "units" along the edges of the rectangle.

How do you address / fix this misconception with 7th-graders? That's the fun part. You can address it by walking over to the kids who have their hands raised (they have to get it "checked off" before they draw it down on their paper), and you count the squared units inside with them. You touch each squared unit inside the rectangle -- fortunately there are only a small number of them -- and you count slowly until the kids can see / are convinced beyond a doubt that their originally constructed area is incorrect. And then you just walk away and let them figure out what they did wrong. Within a minute or two, they'll get it on their own by trying new things with the rubber band -- and they'll be very excited about their own discovery that units, not points, are what matter when we measure objects!!

* On a Geoboard, their mistakes are indeed malleable and a "normal" part of the learning process. Kids don't feel discouraged OR frustrated, which they might if they had to erase and re-draw on paper. You are able to completely normalize mistakes by simply making the process of correcting the mistake fun.

* Geoboard allows kids to experiment. Can they build multiple, different-looking rectangles with the same area? Do those rectangles have the same perimeters? Kids can reach out and touch the perimeter units as they count. This is good for their connection between what's visual and real, and what is written on paper. For me, I try to let kids use a continuum of representation -- tactile, to dots on paper that look a lot like what they use on the Geoboard, to then "abstract" images with only labeled side lengths that are not drawn to scale. The continuum is what will help the developmentally more "concrete" thinkers understand abstract formulas.

* Using a Geoboard, kids can build shapes within other shapes. A classic connection is building triangles within a rectangle. Can they build different ones? Can they convince themselves why the different triangles that can "maximally" fill a rectangle will always have the same area? If I then change the triangle to be something smaller, can they fix the rectangle as to help them find the new triangular area? Those are things that they can explore with their hands and talk through with their partners and with me, and eventually they can deduce the area formula of a triangle. The kids can also build overlapping parallelograms and rectangles, for example, to discover the connection between their area formulas. (Naturally, some of this can be done with cutting and pasting paper, too. I did this with my 9th-graders and they all remembered the various 2-D formulas very well on the test AND could easily apply them. But, Geoboard is nice because within minutes the kids can build multiple examples in order to draw a more confident conclusion.)

* And then, can they extend these discovered concepts to abstract diagrams? And when they do, does it fill them with excitement to be able to make that little connection between a newer, more abstract representation and the concrete one with which they are now familiar with? Geoboard is not the end-all, but it certainly paves way to future learning.

I loooove Geoboards for introducing geometry, because I feel that it fits so well into the adolescent need to explore, make mistakes, and transition from the concrete gradually to the abstract! Why give the kids formulas, when they can learn to develop them on their own?

Addendum March 16, 2012: The shearing activity of triangles was also quite lovely! The kids gave me the "Duh! It's so obvious!" response when I asked them to compare areas of triangles before and after being sheared. (I had them cut the triangles into strips, shear them, and then glue the sheared results back down into their notebooks.) Many of the kids are now able to successfully figure out triangular areas without being told a formula, and are also able to re-draw what the sheared triangles had looked like BEFORE they were sheared, in order to better visualize their area calculations. Lovely! The next follow-up geometry activity will again use the Geoboard. Some of my faster students have already begun working on it. It'll guide them through building squared areas such as 25 and 16, in order to learn to recognize which numbers are not perfect squares, and to be able to still estimate the sizes of those decimal square roots. (This is, of course, in preparation for solving Pythagorean Theorem equations a bit further down the road.) Love teaching Geometry to my 7th-graders!!

Wednesday, March 14, 2012

Where is My Lesson Plan?

I am not a fan of full-fledged lesson plans but I am also not a fan of not making any plans. Normally, I carefully plan out the worksheets and I make answer keys in advance to gauge timing and to anticipate student questions, but I don't write out much else.

Today, I wrote out a detailed lesson plan the way I used to, when I needed to turn in weekly lessons to supervisors or when I shared lessons with others. I wrote it out because I was being formally observed, so I wanted to be proper. But then, I misplaced it DURING the lesson!!! OMG I'm such an idiot. I tried not to look frantic but I really couldn't find it after putting it down somewhere. So, I decided to play it cool and just to keep going, obviously. (What we always do when lessons hit an unexpected turn.)

Naturally -- as these things always go -- I found the printed lesson plan immediately after class. It was under my stack of extra handouts the whole time. I had fully accomplished the aim, and the lesson had gone pretty well (the kids enjoyed and understood the algebra, and they helped each other as usual while I walked around to answer questions and to check in on the weaker students), but I had had to make up check-in questions on the spot since I couldn't find the "real" plan with the pre-thought of questions. sigh. Here is to hoping that my supervisor doesn't care that I didn't actually follow the plan to a tee.

PS. I used the exponent intro lesson for my formal evaluation. It worked very well; it didn't need much introduction but I did briefly introduce the handout by going through a couple of concrete numerical examples with the whole class, just to introduce/review relevant vocabulary words and to reinforce how to interpret the meaning of the coefficient vs. the exponent:

5*5*5 = 5^3 <--- 3 is the "exponent" or "index" (plural: "indices")
5*5*5 + 5*5*5 = 5^3 + 5^3 = 2*5^3 <--- 2 is the "coefficient", and this expression means we have "2 copies of 5^3"
x*x*x*x + x*x*x*x + x*x*x*x = 3*x^4 <--- "3 copies of x^4"

I would highly recommend using this (see link above) as a lesson for introducing exponent rules for the first time!! Kids were awesome and able to articulate the big ideas by the end of the 80 minute lesson. They were either primed for starting division of terms or had already finished that part off as well. Yesss.

Monday, March 12, 2012

Substitution with Some Flair

Some of my students who prefer not to solve systems by substitution have come up with a new way of substitution that I actually REALLY like. I am going to share it here with you.

Say the problem indicates to solve this following system using substitution (I know, it's totally artificial to prescribe a specific method, but for shared assessment reasons, I want them to still be prepared for questions like this on the semester exam):

2x + 3y = 48
3x - 4y = 4

Basically, some of the kids who dislike substitution have invented a hybrid of the two methods of substitution and elimination. They first scale the equations to get matching terms:

(2x + 3y = 48)*3 becomes 6x + 9y = 144
(3x - 4y = 4)*2 becomes 6x - 8y = 8

If you solve the first equation for 6x, you get 6x = -9y + 144. Then, substitute this into the second equation, you get: (-9y + 144) - 8y = 8. Ingenious! I was impressed that they just invented this hybrid method. It's much easier than getting x = -3y/2 + 24 and plugging that into the second equation to get 3(-3y/2 + 24) - 4y = 4, because the hybrid method they have invented bypasses all of the fractions immediately and still satisfies the problem requirement of applying the concept / skill of substitution. Their method of substitution is so much more elegant than our traditional substitution.

LOVE! Amazing what kids can come up with all by themselves.

I am simply bubbling with excitement to finally start new topics this week! My short attention span really gets the better of me.

Encouraging Emails

These below are the nicest emails I've gotten thus far this year from my students' parents, both of them out of the blue. Sometimes when I am in the thick of things like now, I try to think about how much I simply adore my job and how, even if it means I have to go home after a long day and still do a million things just to keep up with the crazy pace of things, it's just such an incredible and humbling journey to see kids grow as people between August and June.

"You have really made a difference this year for [my son]. Before, he was under the impression that math was not his thing and you have been able to make him see that he is capable. As a result he has become much more confident about his math skills! Thank you!"

"This is [so-and-so's] mother and I am writing you because I am especially grateful for your work. I must mention that the time we started at [your school] was very difficult because the school doesn’t have math levels and we felt that his math lessons were lacking in engaging, intereactive and varied, [my son] felt he could give more. So [this year] started with a lot of expectation about Math curriculum and how the school could improve it in order to leverage the children talents not only for our son’s benefit. And I think the school did it because of you. I saw [my son] enjoy Math again and I want to thank you for making my sons math lessons his favorite."

Remind me when I become a parent and I get on the other side, that the best way to help a teacher feel motivated to do a better job is with positive reinforcement!

Sunday, March 11, 2012

Burnt out

I am feeling really burnt out. My goal for this coming week is simply to be good to myself. That means leaving work at 4:30 everyday, if I can possibly swing it.


Addendum Monday March 12, 2013: I couldn't leave at 4:30 today!! I had a kid who stayed for extra help until after 4pm, and after that I still had to make copies and to pack up. BADNESS!!! I didn't get on a bus until 5:24pm, and I actually still need to grade tonight as I have promised my 7th graders they'd get their business plan rough drafts back tomorrow in class. (I had collected those on Friday and they're slow to grade since I check every number.) I need to do better about taking it easy!!!

At least today was a good teaching day. Even the one "difficult kids" class I had to cover at the last minute went pretty well, as evidenced by a girl approaching me to be her tutor. But, damn! I really need to do better with keeping my own promises. Tomorrow and Wednesday I have after school meetings, so I'll have to do my best to be OUT by 4:30. :(

Wednesday, March 7, 2012

Geometry Visualization Tasks

This is a fun fractal pattern I came across yesterday. Questions to ask students:

* How many squares are in stage n? (Can you write both a recursive and an explicit formula?)

* How big is the area of each square in stage n? (Can you write both a recursive and an explicit formula?)

* How big is the total shaded area in stage n?

* How big is the total perimeter in stage n?

* Are these arithmetic or geometric sequence patterns, or neither? Justify your answer.

* At what stage will the total shaded area be less than 1/1000 of the original area?

* Can you design your own square-based fractal and repeat the steps above?

And this is another fun/short geometry problem I came across today.

Find the area of the shaded region, if each circle has radius r and A, B, C are the centers of the three circles.

(And for extra fun, they can also find the areas of the other sections of the Venn D.)

Both of these are attributable to IB resources. The latter came from Mathematics For The International Student: Mathematics SL and the former is a modified version of a task from the MYP Taskbank.

PS. I keep missing my 6:04pm bus, because at around 5:30pm daily I start playing with random math problems. sigh. This is no good, because the bus only comes once every 20 minutes, and even after I get on the bus, it still takes me another 60 minutes to get home. I need an iPhone so that I can get an app called iProcrastinateAboutGoingHome!! ...This also means that I am at work for 11 hours almost daily (not including my one-hour-each-way commute). Work-life balance during year 1 is always a toughie for me, since I have naturally obsessive working tendencies. :(

Addendum March 9, 2012: I was playing around with circular fractals yesterday. If you keep replacing circles with two smaller inscribed circles (whose radii are 1/2 of the original circle), then the total perimeter of the shape stays the same and its total area decreases at each stage by a factor of 2. Neat eh? And not very intuitive! If you repeat this indefinitely, the final "shape" approaches two slightly squiggly (basically straight) lines, and together they still have the same length as the original circumference!

Sunday, March 4, 2012

Geometry in Grade 7!

YESSSS! I am excited, because soon I will get to introduce basic Geometry to my 7th-graders! yay. I've never done this before. I have some ideas based on having taught Grade 9 Geometry a few times now, but how to translate that to the correct middle-school level and to meet my students' interests and spatial understanding is never quite so straight forward.

Here are my ideas so far. I'd love to hear what you think!

* I am going to start with introducing the idea of non-square quadrilaterals. I found and modified a nifty activity that asks students to find as many different quadrilaterals as possible within a 2-by-2 grid. I will start with that, to encourage them to think of Geometry as a discipline of math that requires creative thinking. Following coming up with their own quadrilaterals, they will be introduced to the idea of area being "counting the squares contained" and be asked to find as many areas as they can, for those quadrilaterals. I want the first day to just be a day to gauge their creativity and problem-solving, and if some of them manage to come up with some cool (insightful) area reasoning, I will have kids share them with the class.

* Then, we will move on to area of triangles. Typically I introduce this concept to my 9th-graders by letting them play with the Geoboard, so this year I will do the same for my 7th-graders. They will build triangles within rectangles, to see why area is always 1/2 of the rectangle. From there, we will discuss shearing of triangles (using hands-on cutting/pasting of strips of area), in order to understand how to construct a "nice" triangle that has equivalent area to a sheared triangle. They will then do some triangle mixed practice.

* Then, we will talk about Pythagorean Theorem from a historical perspective, because Pythagorus was such a badass cult leader! Following which, they'll obviously need some mixed practice for a few days, blah blah.

* Based on how the kids do with these introductory topics, I will decide where to go next... I think in the standard Grade 7 curriculum at the school, they just have to be able to do basic area and perimeter, so we'll almost certainly have to cover circles to some extent. (I am less excited about this, as you can tell, because I still need to think about how to teach the formulas not-so-rotely.) But to me, I don't really see why they cannot start working on composite areas relatively soon after that! OR to jump ahead to doing some fun visualization exercises in the 3-D space. :)

* For extra hands-on fun, I am definitely planning to have my students use toothpicks and marshmallows to build tetrahedrons and cubes, to explore/explain why triangular structures are used commonly by architects! Maybe they can even build a toothpick bridge as a nearing-end-of-year project...

Do you have any killer ideas for introductory Geometry activities that are good for 7th-graders? I would LOOOOVE to hear them.


One of the things I think about sometimes is how much learning you do when you teach in different places. Truly, circumstances necessitate adaptation. If I did not move between schools, I think that I might have stagnated after a few years.

In my first school, I learned...

* write scaffolded lesson plans in lieu of lecturing, because I didn't command enough respect from my first-year students for even 15 minutes at a time to complete a mini lesson.

* give up control of my lesson-planning process and to compromise with another partner with whom I shared co-planned lessons. Some days this was really great, and other days the results were awful. In either case, we learned to compromise, to communicate, and to reflect on how we can improve. It also forced me, in many cases, to step beyond my comfort zone and to try out someone else's lesson format (which in the long run, was good for me).

* ...that what we do makes a difference. What I saw was the transformation of inner-city children between grades 8 and 10, between the first time I taught them and when I taught them again. I learned that the greatest gift I can give a child is persistence, because with that they will work through everything.

* not rely on textbooks. We didn't have/believe in textbooks. I learned to read standards and state exams and to turn them into objective outlines, pacing guides, project brainstorms, and ultimately teaching material. I learned to think based on my intuition and not based on something that is already written in front of me, which even now is how I begin every unit. I find that I create things that are better suited for my students when I start working in a vaccuum and only afterwards look into supplemental resources.

* admire other teachers' work. The school was so amazing and so hands-on, that if you are humble enough to pop into classes, learning opportunity was everywhere. There are things that I had only heard about or seen briefly at that school that I have since adopted for my own classes.

At my second school, I learned...

* teach with the freedom of having no rigid written curriculum and no standardized test pressure. In the lack of those two things, there is limitless possibility.

* switch between using and not using textbooks, depending on how good I think the written material is. What freedom to have!

* ask to borrow resources and to look into dusty resources on the shelves that no one has used in years. I learned to network with other people (science teachers, librarians, resource assistants) to help me write lessons and to investigate resources for me.

* utilize the amazing year-round weather to take my students outside for outdoors lessons.

* write GeoGebra-based investigations to take advantage of the easily accessible computer lab.

* communicate with all parents weekly over email.

I am now at my third school. This school is fast-paced and my colleagues are vastly more experienced than me in their areas of expertise. One of the challenges that I face is how to carve out my own space for effective teaching without overstepping the pacing and assessment framework that we all share. One of the things that I find that has benefited me the most in this new environment is to be able/comfortable enough to reach out and ask tons of questions, which is something that I have learned from my earlier schools. It has already forged some amazing working relationships with some of my colleagues, and helped ensure that this transition into a new curriculum was as smooth as can be.

I am grateful for my opportunities, and I can only hope that this road will always lead to greater challenges and greater rewards!

Saturday, March 3, 2012

Going Sloooooooow on Exponent Rules

One thing that always troubles me is that many students tend to want to rush to draw procedural generalizations before they reach a solid conceptual understanding. What this means is that in two weeks, when I'm not standing in front of them, they do not remember how to properly apply the rules and cannot even retrieve the relevant concepts to re-engineer those rules! argh.

This year, for teaching exponents, I am going to sloooooooow them down to try to avoid that.

Here is my attempt at pulling together a worksheet on exponents (which, granted, isn't the most exciting of topics). Check it out - I like this worksheet and think it will work pretty well, even though everything in it looks very basic. We need to build sloooow conceptual understanding, and after that we will drill the rules using some games!

Addendum March 14, 2012: This lesson (parts 1 and 2) worked like a charm today!! I am feeling extremely positive about the outcome. The students could explain to me with NO PROBS WHATSOEVER why when you add or subtract terms with exponents, the resulting exponents don't change, but when you multiply them, they do change. My faster students were finished with the entire worksheet and did not have any misunderstanding at the end. They're ready for Phase 2: simplification drill games!!