## Wednesday, March 30, 2011

### Great Expectations

After having recently read some articles about improving students' algebra readiness at all levels of mathematics, I've been thinking about how I could incorporate that into my Geometry classes. Fortunately, we're on a proofs unit, and what better way to introduce heavy-duty algebra than to ask them to prove simple theorems in the coordinate plane -- theorems that they already are familiar with?

--Well, I sort of went overboard on this, to be honest. I quite possibly overshot what can be considered reasonable expectations for regular 9th-graders. But, I was so impressed by my 9th-graders. I gave every student of mine -- honors and regular -- a very algebraically rigorous proof. I decided that of course I'd have to guide them through it by demonstrating a couple of proof parts on the board and then letting them duplicate / struggle through the super long algebra process for the rest of it. It took us two whole classes to do this one proof, but I think it was super worth it.

So, here we go. Student work samples to come, but I am so excited by this worthwhile task that I couldn't wait for those work samples to come in to post about it!! (My regular kids are turning in a draft of their "proof" next class after we fix it up a little and tie in explanations for each step. Their homework tonight is to re-write the proof draft neatly, step-by-step, so that I have a shot at following and verifying everyone's algebra work.) I'm really excited! I had one middle-of-the-road kid in one of my regular classes tell me today that she likes this. I believe she said something about it being really "deep." It made me very happy, and I told her that math in college is all like this -- no numbers! Just variables!

So, here's the prompt:
Prove that if you start with a triangle with coordinates M(0,0), N(a,b), and P(c,d) and you construct "midsegments" going from one midpoint to another, then each of your midsegments will be parallel to one original edge in the triangle and have only half of the length of that original segment. Assume that a, b, c, and d are all nonzero values.

To balance between letting them try it and guiding them along, I first let them write the goal and the given and to attempt to draw the diagram. Then, as a class we fixed our diagrams so that the midpoints X, Y, Z are all labeled the same way on everyone's paper (so that everyone has segments XY and PM being parallel, for example). I drew this on the board to help guide our discussion:

Then came the intense mini lesson!!

I showed them, step-by-step, an example for how to show using algebra that one midsegment is parallel to an existing edge in the triangle:
$Location\hspace{1mm}of\hspace{1mm}X=(\frac{a-0}{2}, \frac{b-0}{2}) =(\frac{a}{2}, \frac{b}{2})\\\\ \indent Location\hspace{1mm}of\hspace{1mm}Y=(\frac{a+c}{2}, \frac{b+d}{2})\\\\ \indent Slope\hspace{1mm}of\hspace{1mm}\overline{PM} =\frac{rise}{run} = \frac{y_2-y_1}{x_2-x_1} = \frac{d - 0}{c - 0} = \frac{d}{c}\\\\ \indent Slope\hspace{1mm}of\hspace{1mm}\overline{XY} =\frac{rise}{run} = \frac{y_2-y_1}{x_2-x_1} = \frac{(b+d)/2 - b/2}{(a+c)/2 - a/2} = \frac{d/2}{c/2} = \frac{d}{2}\cdot\frac{2}{c} = \frac{d}{c}$

For Day 1, all they worked on the rest of the class (after our mini lesson) was to show that the other two midsegments are also each parallel to an edge in the original triangle. (Believe me, it takes that long for a whole group of kids to each crank through this algebra, since they needed to do it on their own twice, once for each remaining midsegment. If you look closely, for Day 1 alone they would need to be able to algebraically find midpoint and slope, and to correctly add/reduce/divide algebraic fractions.)

Then, on Day 2 (today), I showed the regular classes how to prove using algebra that the midsegment is 1/2 of the length of that original edge (which again involves some hairy algebra, including distance formula, square root simplification, and some fractional operations):
$length\hspace{1mm}of\hspace{1mm}\overline{PM} = \sqrt{rise^2 + run^2} = \sqrt{(d-0)^2 + (c-0)^2} = \sqrt{d^2 + c^2}\\\\ \indent length\hspace{1mm}of\hspace{1mm}\overline{XY} = \sqrt{rise^2 + run^2} = \sqrt{(\frac{b+d}{2} - \frac{b}{2})^2 + (\frac{a+c}{2} - \frac{a}{2})^2} \\ \indent = \sqrt{(\frac{d}{2})^2 + (\frac{c}{2})^2} = \sqrt{\frac{d^2}{4} + \frac{c^2}{4}}= \sqrt{\frac{d^2 + c^2}{4}} = \frac{\sqrt{d^2 + c^2}}{\sqrt{4}} = \frac{\sqrt{d^2 + c^2}}{2}$

Then, just as before, the class had to repeat that process for the other two midsegments. Some of the kids were expectedly getting lost in the algebra, but most of them were able to re-find their ways with some help from their partners, and by the end of the class just about everyone was done with all of the algebra work that is required to complete this proof!

I am SO happy! These are my regular Geometry kids!! It just goes to show, I guess, that I need to dream up what's (im)possible, in order to challenge and to allow my kids to rise to those heights!

PS. I'm trying the new LaTeX in Blogger thing. I think it works ok but the indentation is all messed up. What gives? I had to insert extra indentations in order to make it all line up.

PPS. Um, one kid did cry yesterday. I'm not sure why, to be honest, because when I checked in with her group 5 minutes before she had cried, she was still at a "working-level frustration." Then, she went from zero to crying in 5 minutes. Today she was doing much better, and she got through all the algebra parts in the end, with time to spare! yay. I told her that in the future, she should raise her hand if she feels like crying........... sigh. That's a record-breaker for me, that's for sure.....

Addendum 4/11/11: One exemplary student work sample here.

## Tuesday, March 29, 2011

### Letter to my (Select Few) Students

I wrote an email to a few of my juniors. My expectations for them and their work habits are going up by the day, and I fully expect them to rise to the occasion. I started a "weekend homework assignment" thing for Q4, where every weekend I will assign them an extra homework assignment that takes them at least 1 hour to complete. These assignments have to do with old topics, so it's extra taxing because they have to struggle to remember details of what we did in Q1, and to read the textbook and whatnot in order to jog their memory. This is an effort to get them to be more college-ready, in terms of habits if not in terms of content. On Sundays, they have to send me questions by 10am, and I promise to then respond promptly. On Mondays, I collect and grade their "weekend assignment" based on 2 categories -- "weekend effort" (Check, Check plus, Check minus) and also accuracy (Check, check plus, check minus), and I circle and carefully annotate every problem that they missed. Took me a long time to do it this week -- 3 hours for about 10 kids' assignments. I'm hoping I'll have the perseverence to last through all of Q4 this way. --Anyway, they then get to take the assignments I've graded/returned, and to correct their errors and to resubmit by Friday for a better grade.

Anyway, here's an email I wrote to my few kids who failed to turn in the first weekend assignment. Not surprisingly, despite my many reminders they came back with various excuses on Monday.
Dear students,

I am writing because either you did not turn in a weekend assignment on Monday or you turned in one of very poor quality that I could not really begin to grade it.

I would like you to seriously consider making up this assignment, and doing it to the best of your ability (including asking for lots of help). I make no apology that it's going to take you well over an hour, if you're doing it right. But, it'll give you the opportunity to get detailed feedback on Ch. 1 material that will be on the final exam. Believe me, these opportunities will come and go every week, and you will regret in June that you weren't all along asking me questions about these past topics throughout Q4 so that you could understand them piecewise.

Looking forward to you being responsible,
Ms. Yang

Maybe the email will have no effect, but I refuse to let the kids think that this is just some other missed assignment, because it's so NOT. To me, it's about much more than that. Like I said in the email, it's about them showing the right mindset for being college-bound students.

## Monday, March 28, 2011

### Top-Down Approach to Proofs

I am trying a new approach to teaching proofs this year. To give some background, last year teaching proofs was awful. We basically worked from the bottom up: we took each of the unthinkably boring triangle congruence theorems and learned them separately (doing hands-on investigatory blah blah and then some basic textbook practice), which took us well over a week. And then we finally started doing proofs, which I did by giving them pieces of incomplete proofs and asking them to fill in the missing pieces using theorems from memory, like our textbook had recommended. Kids absolutely hated it for two reasons:
1. the learning was drawn out (for both the strugglers and the high-achievers),

2. the proofs they were able to do in entirety were visually "obvious", therefore they lacked basic motivation to do them.
I hated it as well, because I was struggling with teaching kids basic theorems, proof construction, and even basic geometry naming convention and notations all at once! It was uber-frustrating and ineffective.

This year, I sort of said, "Screw all that." The proofs can wait until the end of the year, until we've more or less mastered most basic geometric calculations and formulas and notations. Because if we wait until the end of the year, then
1. I can spend some extra time on proofs if I need to, without worrying about running out of time for other topics,

2. they will already have a more solid foundation for visualization and reasoning, on which we can build their proof reasoning,

3. their geometric vocabulary and ability to name angles / segments precisely will be ready as well.

4. we can work on "juicier" proofs, which eliminates the motivation issues I saw before.

...So, here we are in Q4 and we're just getting to proofs now. This year, I also made a huge decision that maybe memorization of all the base theorems isn't so important. For my honors kids -- yes, they'll still need to memorize most of them before the test. But for my regular kids? Is it really important that they can pull "Reflexive property of congruence" off of the top of their head, instead of pulling it off of a list of theorems? The answer, I decided, is no. (I have the blessing of my students not needing to take a standardized Geometry test down the road.) I decided that the true worth of doing the proofs is that they get to practice the formality of their reasoning techniques, so I made some huge strategic changes:
1. We don't do the two-column format. Every proof still needs to start off with a "Given" and a "Goal" and a diagram, and they still need to write a numbered list of statements, each accompanied by a reason. But they don't need to do the two-column thing, which never made much sense to the kids anyway. Instead, I make them go through each completed proof to color-code their reason and their statement separately, in order to show that each statement in their sequential logic is fully substantiated with a justification, right there within the same numbered step! The kids find this to be very natural. So far I have not had a single kid complain about the requirement of a stepwise justification being confusing, in my regular or honors classes! (In fact, some of the artistically inclined kids can hardly wait to color code their proof at the end.)

2. They get a reference sheet of basic theorems to use, for now and on the test. (Well, regular kids do, anyway. Honors kids will have to learn them before the test.) So far, I've been extremely pleased with the tremendous difference I've seen in their ability to construct proofs! Instead of feeling frustrated each time they need to look up a theorem in the textbook like looking for a needle in the haystack, having a reference sheet means that they can quickly look it up and feel very confident when they know that it's the next necessary building block in their logical sequence. We've just started this late last week, and it's going very well. I think that toward the end of the week, I'm going to switch out their diagrammed reference sheets with a flat list of theorems with definitions, and then sometime next week I'll further remove the definitions so that by the time of their next formal assessments, they're used to only referring to a list of theorem names -- no definitions or descriptions will accompany those theorem names. Slowly removing the scaffold will, I believe, help keep the focus on the logical process, while building their independence and their ability to recall basic theorems...

3. I'm not going over why SSS and SAS and ASA and HL theorems work. I'm sorry, but if I had to teach each of them separately again, either I'd kill myself or my kids would kill me, out of sheer boredom. I think that a smart 9th-grader who stops to think about those theorems can figure out for the most part why they work, and a struggling 9th-grader won't remember anyway; we don't need to have everyone suffer through the tedium of those lessons.

4. No more "fill-in-the-missing-step" proofs! They're kind of pointless, in my opinion. Kids can always fill in a blank. That doesn't mean they know the first thing about writing proofs. This year, we write every proof from scratch, right from the beginning.

Anyway. Just thought I'd share my approach. This whole top-down approach to proofs is making it a whole lot less of a nightmare for me. We've worked on some triangle similarity and congruence proofs so far, and we're going into circle proofs and coordinate plane proofs next. Kids are doing fabulously with looking through those reference sheets and putting logical steps in order and justifying each step... and I think they even think they're kind of fun (like a puzzle). YAY!

Addendum 3/29/11: Since a couple of you requested, here's a take-home proof that I assigned a "fake grade" to, to informally assess the kids (and so that they can see what grade they would have gotten, had it been a real quiz or test). You can see the color-coding for statements vs. reasons. If you are looking closely at the proofs, you should know that I crossed out angle and segment notations where they weren't naming the points in the correct order (for example, saying angles ADE and EBC are congruent. Yes they're referring to the correct angles, but no they should have said angles ADE and CBE instead). For those details, I took off only 10% total in their "fake points", but I corrected them everywhere on their paper so that they'd start to pay attention. In other words, if you look at this first proof below, that girl's proof is splendid besides that minor issue, even though it looks like I marked it up a bunch. --And let's keep in mind that these are my regular Geometry kids, only 2 or so days after we started doing proofs! :)

## Saturday, March 26, 2011

I can't believe I am saying this, but I will only have a few more "Life in El Salvador" posts left to write this year. The end of our time in El Salvador is coming so soon! It's really quite amazing how quickly June is approaching. So, here are some updates before I forget what cool things we had done in this beautiful country.

Somewhat recently, a bunch of us went on a night tour to the Cemetery of the Illustrious. The place was huuuge and had a lot of really ornate graves and tombstones.

One of the most interesting headstones was this unlabeled one of an old fascist president, Maximiliano Martinez, who had caused the slaughter of 30,000 natives during his dictatorship and had chosen to have a nameless tombstone in fear of post-mortem retaliation.

To fully appreciate how special these "illustrious" tombs are, I think you have to understand what "normal" Central American cemeteries look like. Below are some pictures we took of a regular cemetery, which we had walked past during a hike in February. These local cemeteries always have a lot of colorful flowers surrounding the tombs, and they are far from the gothic feel of the Cemetery of the Illustrious.

Here are a couple of other pictures from the February hike.

The hike itself was long (maybe 6 hours up and 3 hours down?) and not totally scenic. Pretty uneventful, I'd say, except that you come across trenches that are remnants of the Salvadoran Civil War (which ended in 1992, very recently), and in the end you also come across this terrible sign, warning of land mines ahead:

The hiking guides told us that after the war, Belgian experts had to be brought in to remove the mines with special equipment. Not just on this mountain, but on other nearby mountains as well. It serves as a reminder of just how recently the war had ravaged this country.

Then, most recently (today), I tagged along with some friends to whale-watch in Los Cobanos! I like Los Cobanos a lot, because it's a deserted beach with a chill hostel (whose owner is a great hostess and an awesome singer) and there is a food hut right around the corner from the hostel that serves delicious grilled fish, stuffed with cilantro, garlic, peppers, and salt. YUM.

Anyway, we stayed overnight last night to hang out at the beach, and this morning we woke up bright and early to go whale-watching. Mission accomplished!!

If you can't tell from the pictures above, there were two whales that we saw -- a mom and a calf. They were swimming around us peacefully for about 5 to 10 minutes, periodically sticking their backs/fins out of the water -- sometimes at the same time! In the end, the mom whipped her tail into the air. It was beautiful!

On the same trip, we also saw a variety of other wild life -- (large black) dolfins, a sea turtle, various jumping tuna, and birds! We also got to swim for a good while, and it was SO NICE to hang out in the warm ocean water.

...That's all the updates for now. :) Our next big trip will be to Roatan, in Honduras. It'll very likely be our last international trip before we move!! Que lastima!

## Thursday, March 24, 2011

### Weak Verbal Skills in Math

I was thinking the other day about how I ought to sit down with a couple of my 9th-graders to have an INTERVENTION on how to read test instructions. Like, maybe make them take a highlighter and highlight every verb in a question taken from an old test, so that they can make sure that they're doing all parts of the actions required. "Find ... Label ... Justify... Round..." I wondered: Are they even seeing all of those verbs and considering each one as a separate requirement of the problem?? Because the same kids that seem to never read instructions don't appear to be getting any better at it over time, and they're losing significant amounts of points every assessment because of it. --So frustrating! Argh.

Today, I got a chance to try this out on one kid who came to see me after school for help. She actually came to see me about content help in looking over a recent quiz, but I told her right away that I think that maybe part of her issue is related to her ability to interpret the instructions. So, I sat down with her and asked her to highlight the verbs in the first problem from our last quiz. To my surprise, she immediately highlighted a verb-turned-adjective in "the labeled point ..." and missed all of the important imperative verbs that actually tell her what to do! Oh, boy. I had suspected that her verbal skills were affecting her performance in math, but I had no idea to what extent until I pointed at a word she had missed highlighting, "justify", and asked her if she knew what it meant, and she said No!! Yikes!! (Especially scary because I think I use the word "justify" everyday in class, and almost certainly on every assessment.)

Of course, I explained the word to her and explained that it's OK -- and mandatory!! -- that she raises her hand to ask for clarifications on directions during a quiz/test. I also reviewed with her the idea that most key verbs occur in the beginning of a sentence, if the sentence is trying to tell you what to do. She actually knew the majority of the material on the quiz that we were looking over, but had gotten a crummy grade because she didn't follow most of the instructions (because she had interpreted them erroneously). For example, Problem 1 had asked her to determine whether an already-labeled point was either the circumcenter or the centroid of a triangle, and to justify her choice by describing the special properties of each vocab term. She had chosen the correct term, but her description went on to be very vague and failed to point out the differences between the two types of concurrency points. Part of the issue? She didn't understand that she had to support her choice by stating the differences between the two points -- because she didn't know what "justifying" meant!

Holy smack. It's making me consider calling my other struggling/verbally questionable kids in one by one, so that I can individually workshop them on reading instructions. Have you done this type of intervention before? What other strategies can I give them besides highlighting the verbs? (I don't want to give them strategies that are too time-consuming. Highlighting seems like a reasonably easy thing to do, even given limited time.)

sigh. If only I had thought of doing this earlier in the year, I wouldn't be here kicking myself now in Q4. I guess it's one of those things that you live and learn. And this is why it is absolutely critical for us to be working with our kids on verbal skills across the board. If your kids have poor verbal skills, do you think they can correctly interpret what "Segment AB bisects Segment CD" means? I said the words "subject" and "object" to my kids today in reference to that mathematical statement, and half the class looked at me blankly. I had to rephrase, "CD is the one that's receiving the action. It's the one that's being cut into 2 equal parts." Believe it or not, we are back to Grammar 101, in Geometry class, in order to move on to construct proofs correctly.

### Student Apathy

One thing that's really disheartening for me every year is that there are always a few kids who refuse to put in anything extra into the class. They show up, they'll do whatever you tell them to do in class, and then they leave and then NOTHING. Their mathematical brain shuts off on the weekends and they don't do anything, and they think you are crazy for expecting them to retain anything from Friday, on Monday. If you have a quiz on a Monday, they'll just take a zero, because the alternative is that they'd have to do something at home on Sunday to review, and that simply is not going to happen.

That's super disheartening, because it makes me feel like I am the only one that cares about their learning or their grades, and that I'm fighting the fight alone. I think they think that it's easy for teachers to have faith and to keep wanting the best out of them. If only they knew how deeply affected we are by their apathy.

## Tuesday, March 22, 2011

### Why Teachers Like Me Support Unions

Today is March 22, 2011. In response to all the craziness going on around the States, edubloggers have been called upon to submit a piece today to explain why we personally support unions, in the spirit of Edu-Solidarity. Here are my thoughts:

I wasn't originally trained as a teacher. By the time I graduated from college, I had an engineering contract in my hand that said, "$75,000 starting annual salary, with$10,000 sign-on bonus each of the first two years." I am not trying to brag, but I am putting those figures out there so that you know that I wasn't exactly aching for money when I decided, a few years later, that I wanted to teach math to kids in public schools.

Why does a teacher like me support unions? I support unions because I looked around at the age of 24, and I realized that if I went to California to teach, my salary would drop to be about \$25,000, give or take. At the time, given my existing situation, that number seemed utterly unacceptable. I wasn't going into teaching for the money, but the percent loss I would have in pay seemed utterly unreasonable to me at the time. It made me almost give up the idea of teaching altogether.

In the end, I joined the NYC Teaching Fellows and moved to NYC, where the union is strong and the pay is better. Still a significant paycut, but one that I was prepared to make in order to give my dream a shot. --And I am so glad I did, because I would never have known what I would have missed out on. I am yet an inexperienced teacher, but I do not go through a single day without thinking about my job and worrying about my kids as though they were my own.

Without strong unions, we would not have the decent working conditions that make teaching seem like an attractive (or even viable) option to qualified candidates. --Let's not even talk about how the kids would be impacted, if they could only have disgruntled teachers who had chosen teaching only in lack of better career options! If kids are our future, then the teacher unions must be our present focus in order to fully support that future.

## Monday, March 21, 2011

### My Vision of an Ideal School

I've been thinking about this ever since I saw (I think this is old, but I only recently came across this) Sir Ken Robinson's illustrated TEDtalk about modern schools as a remnant of the industrial revolution. It makes a lot of sense what Sir Robinson's saying about our schools having a real "factory" feel to them, and in a way, it hits home for me more so than some of his other TEDtalks. This particular talk has helped me come to realize that I'm not a fan of the whole bell system, and actually have never been.

--To give some background, I consider myself a pretty decent student. I genuinely like learning all subjects, and have always done reasonably well in school with some effort. But, most of the time the lectures are too slow for me. In high school, I learned to distract myself daily with other things in between the first minute the teacher's talking and the fifteenth minute he's still talking about the same thing -- especially in subjects like math and science, where there wasn't much encouragement for analyzing nuances at the high school level. In high school, we didn't have much choice but to sit through all 50 or 60 minutes of class each period each day, and I did my best not to be rude to the teachers or my peers, even though my mind was often a thousand miles away.

As soon as I hit college and I realized that 1. professors were still lecturing at the same boring pace, thereby covering in class only 25% of what you're supposed to learn before the midterm (and expecting you to read the text for the remaining 75%), 2. they don't take roll in classes of size 200 or 350, I stopped going to classes. It seemed much more efficient to stay at home, do all my work, and read through the chapters on my own, hitting up office hours only occasionally with questions, than to sit through dreadfully boring classes that weren't teaching me most of what I needed to know anyway.

In hindsight, it is nothing I'm proud of. I think it's a poor show of my priorities and my character to slip out of class simply because I could and because I felt like it. My grades in college were decent, but by no means were they great. I was doing pretty OK for a couple of years in studying on my own, until I started to get really busy with extra-curricular things on the side, and reading textbooks at home became a chore.

But, thinking back, even though I am obviously responsible for my own choices in college (and I would never say otherwise), I now understand that the factory mode of learning never did work for me. I was a decent student and I took advantage of the liberties in college to do other things that I wanted to do with the time that I had -- but that was a more natural way of learning, and I can now comfortably open up books and a webpage and manage to learn whatever it is I want to learn. Why can't our students be encouraged to do the same? Why can't high school work on a full-time drop-in schedule, where each day each student needs to check in with their teachers to 1. show them work that has been completed, 2. to get clarifications or to do an on-the-spot assessment, 3. to get started on the "next step"? I don't need all 70 of my kids to sit with me for an hour each day, if they are able to sit somewhere else and do the work and come back to pass an oral or written assessment after a few days. For the kids who need to sit with me for two hours a day, I don't want them to feel pressured to leave after just one hour. I also don't need every kid to be moving at exactly the same pace. I certainly don't need them all to be sitting in the same space, confined to their seats for 60 minutes each time. Those are unnatural working conditions (imagine if every non-academic work place had a "bell schedule" that released you for the bathroom every 60 minutes... I think you'd be hearing lawsuits!), so why would we think that these conditions are conducive to our children's learning and growth as diverse individuals? And, how are these conditions helping to build the kids' own sense of time- and priority- and progress-management, which they desperately need to develop in order to be a successful adult??

I think that models like this and this (Thanks, Google Reader!) are the closest to what I am imagining for an ideal school setting, both in terms of learning methodology and physical organization. In my ideal school, kids would have to cover a common set of learning objectives in each subject by a certain time, but if they are advanced, they should be flying right ahead and if they are remedial, they should be given realistic goals of mastery that they need to be accomplishing within the year. Teachers would serve as advisors, and kids would be expected to use other resources such as internet and videos and various textbooks in order to aid their learning. In my ideal school, kids would spend a part of their days working on an independent project of their choice -- for example, if they like to dance, they should be dancing everyday and be working on some sort of choreography that they design, maybe bringing it to a production level with a team of other kids and managing all of the logistical and financial side of the show as well. If they like to make music, then they should be developing a musical expertise and learning about the technology of recording, and maybe tying it into building a music website. The only requirement is that they have to set specific goals in their areas of expertise and to work on reaching them in a concrete, timely fashion, and to stick with a "project" for a predefined amount of time without giving up. This would teach kids persistence and resourcefulness in face of obstacles, when working on something that is meaningful to them. It would also teach them time-management -- if you want time to work on your "independent project", then you need to first put in the time to get your core learning done in other areas, whatever it takes. It would alleviate the sense that kids are just "doing time" inside each class. You don't have to spend 50 minutes on each class each day, if you can manage to master the material in less time!! If a kid's passion really lies in math, they should go for it at full speed and not let the artificial year-long curricula hold them back. I firmly believe that a sharp kid can easily do most standard math courses in 3/4 of the time, so that they can comfortably "skip ahead" without devastating their learning in one particular class (a la the "Geometry summer school"). The only reason why any kid goes to summer school now is so that we can manage to squeeze them back into the production line of learning next year -- and that simply isn't a good enough reason!!

If we want our kids to genuinely care about school and to become active participants, then maybe appealing to their multiple-sensory learning inside the factory-model classroom isn't an adequate or viable solution. Like Sir Robinson has suggested, maybe what we need is a complete overhaul of the system, one that address our students as individuals rather than as a pack.

What benefits does the traditional bell schedule have that I am not considering? I would be intrigued to hear what you think. If the only advantage is group work and planned exploratory activities, then that's not a good one. We can easily have kids sign up for time slots to come see you at the same time on a given day, in order to make the group work thing happen. And most exploratory activities can happen independently, as long as you have the resources set up. (Again, it's just a matter of logistics.) Another possible advantage I see for the traditional bell schedule is "So the teacher won't go nuts from having to 'advise' Precalc and Geometry and Algebra kids all at once to a bunch of kids at the same time!" But, that seems easy. Either you carve the office hour time slots each day into chunks by subject, OR you divy up the subjects by teacher, so that each teacher is responsible for only one curriculum on a given day or in a given week. This could also potentially give teachers within the same department more collaboration and co-teaching opportunities, as it would be easier to rotate around and help out with different classes, if the kids are only coming by for drop-in help anyway. The concept of a "cohort" would not exist as we know it, and you would not view your classes as X groups of Y students, but as X*Y individuals, each doing the best they can so that they can get to the things that they really want to do in school and in life.

## Sunday, March 20, 2011

Here's an old fable that I think my friend Paul once told me:

The other day while I was telling my Honors Geometry class to go home and finish a non-integer circumcenter problem, I said, "When you get stuck trying to find the intersection of two lines using algebra, google how to do it! Look it up on the internet." I was only half-serious. I knew that the next day they'd more or less all come back and act all helpless.

But, to my surprise, the next day, 2 kids came back with solutions! They had googled it! I was happy (but not as happy as I would have been, had more kids decided to follow my suggestion to be resourceful), and I publicly acknowledged their efforts in class. I wanted the other kids to see that it's a GOOD thing that those kids took the initiative to look up how to do something on the internet. It got me thinking: Why do I not have the same expectations for all of my students? Surely if I start having those expectations all the time, they'd rise to the occasion -- in the same way that they bring calculators, notebooks, and pencils to class because it's expected -- and it would also help to shift us away from the teacher-being-the-sole-knowledge-dispenser paradigm.

In the future, I think that everytime I need to review some old rote skill (such as fractional arithmetic), I am going to first ask the kids to go home and to look it up. Because they need to be practicing being resourceful!!

## Saturday, March 19, 2011

### Intro to Instantaneous Rates

I've been getting really good teaching mileage out of my Pringles cannon video. First, I taught my H. Geometry students how to graph that parabola in the calculator and how to use the graphing calc to find and analyze the maximum. (Since I had to teach everything from scratch to my Precalc kids this year, I'm extra cognizant that if I can squeeze extra algebra/calculator skills into earlier math classes in context, the better off they'll be in future years. Plus, it was actually a video from an activity they did in class, so I figured it's as non-pseudocontextual as it gets.)

They loved it! So far, they can use their calculator to: graph a function (or multiple), find intersection of graphs, find min/max, interpret the min/max given a simple situation, and adjust window ranges. Not a bad basis to have before going into Alg2 next year. We've also reviewed fractional math and reviewed solving systems of equations in context of finding non-integer circumcenters, so I feel like I'm doing an OK job preparing them for the algebra that is to come. For these kids, I also hope to get to maximizing volume problems before the end of the year, in between teaching them proofs and doing 3-D stuff, which I'll be doing with all of my regular students as well.

Anyway, on Friday I introduced the idea of instantaneous and average rates to my Precalc kids using the same video*. The Do Now was a simple review of speed and acceleration terms and making tables of values based on a simple description. (As usual, I was surprised that some of them couldn't do a problem like, "What is the average acceleration per second for a car that starts off at rest and reaches 60mph in 3 seconds?") Then, I reviewed with them how to set up the quadratic equation for projectiles before watching the Pringles cannon video*. They noted that the ball was in the air between t=4 and t=9, so I had them graph the function H(t) = -4.9(t-4)(t-9) in their calculator, and they then used the tables in the graphing calculator to fill out this table below. For t values, they used t=4, t=4.5, t=5, etc. ...going up in half-second increments, all the way through t=9. I had to guide them through how to do the averages, while they worked in pairs.

Once they filled out the tables, natural questions arose as they compared their results for the last two columns. I went around to probe why each pair of kids thought that the half-second "speed" (I guess I should have said velocity, to be more precise) is decreasing and becomes negative in the right-most column. They were excited to be able to figure out why all on their own! It's physics in action!! And then, as a class we discussed why the "average speed since launch" actually becomes zero the moment that you land on the ground. One kid raised his hand and explained to the class that it's because your overall change is zero at that point, so therefore your average speed is zero. --Awesome!! In our discussions, I used the term "instantaneous" loosely to describe what's been happening in the last 0.5 seconds (4th column), to distinguish it from what's been happening since the beginning (3rd column).

It was awesome. Later on I drew a graph on the board like this and asked them to describe the "speeds" between different points AB, CD, and AD. The class comfortably told me that AB has positive speed, CD has negative speed, and AD has zero speed. Good conceptual basis for next week, when we move into the actual algebra of finding instantaneous and average rates! I hope that this activity will "stick" with them, so that when they look at graphs in the future to analyze rates, it won't just seem like some abstract concept.

* Most of them had built and shot their own Pringles cannons in physics this year, so I felt like it was OK to just use the video instead of wasting time going outside to do the same thing again.

PS. I suppose I should mention that Obama is coming to El Salvador. School's not in session on Tuesday and Wednesday for street closure reasons. Selfishly, I feel a bit of relief that this comes 2 days before quarter 3 grades are due, even though it means that there is some instruction time lost, obviously.

## Friday, March 18, 2011

### Math Writing Project Brainstorm

In keeping with the theme of reinforcing language skills in all classes, I have been brainstorming ways to implement a writing project in Geometry this year. The only real research/writing project I have ever assigned was a long time ago, and even though some of them had hated the idea of writing in a math class, I was adamant that they needed to be practicing research and writing skills in all classes -- including mine. I think it's about time to do another writing project with my current group of students, but I just need to figure out how.

A history teacher at our school recently did a scrapbook project, the results of which are beautiful and are on display at the library at the school. Here's the description of the project, as sent out by our awesome head librarian:
"After reading historical fiction novels, tenth grade Latin American History students crafted together scrapbooks that included a historical analysis of the novel, student artwork based on the novel, creative writing, documentary evidence, and literary analysis."

I went and took a peek at those projects. Each page was beautifully pop-uppy and 3-D, with tons of student writing to boot! Can we do something like this for math? Can I make my kids research about fractals, Pythagoras, or trigonometry (both the history and the modern applications) and to write about it and to illustrate their work? What if we made a single-issue Geometry magazine as a class? (I have at least one kid I know who is in Journalism and is supposed to be a good editor.) Would we have enough non-duplicate material to include in order to making this a success?

Thoughts? Have you ever done something like this in your classes?

## Thursday, March 17, 2011

### Flexibility

If there was one thing that I wish I could do better with inside the classroom, it's to bring in the sense of play into more lessons. How do you do that consistently? Math, to me, is such a beautiful subject, because it's mostly a series of puzzles, one wrapped inside another. In order to solve those puzzles, you need to have some pertinent skills and some base knowledge, and to be able to use them flexibly.

But, how do I introduce that sense of play on a daily basis? I can't help but feel that I should be doing more of it in Geometry, since Geometry is such a visual subject.

I've noticed time and again that there are problems that showcase how even many of my honors Geometry kids lack basic intuition when looking at a diagram, and they end up way over-complicating the situation. Take a look at the following problem, given on my most recent H. Geometry exam. The directions were deceptively simple -- to find the area of the concave quadrilateral ABCD.

I made this problem and envisioned that many of the kids would quickly get it solved, after our foray into quadrilateral areas. To hint at the fact that they didn't need to cut ABCD further into smaller parts, I even led into the problem with, "Given that Area of ABCD + Area of ADC = Area of ABC, find the area of Quadrilateral ABCD." ...A dead giveaway?!

Anyway, perhaps predictably, some of the kids struggled on this problem. (To their credit, many others did brilliantly, including a girl who was absent for several days during the unit and had to catch up belatedly on all of the heavy-duty trig content.) Mostly because those kids made some false assumptions, such as assuming that Segment BD would bisect Angle ABC. But, even some others who were able to solve the problem correctly took some detours to get there, such as cutting ABCD up into a right triangle and a scalene triangle. It's clear to me that most of them still lack the ability to zoom in and zoom back out on a diagram -- which is what Geometry is all about!!

I've certainly given them "similar" problems to struggle through in class, but especially as honors students, I also expect them to have a level of ability to apply that knowledge to new situations, on the day of a test. But, how do I teach that flexibility?

Anyway, I really feel like I should be doing a better job helping them develop a better sense of spatial intuition. I just don't know how. :(

## Tuesday, March 15, 2011

I'm advising kids on courses to take for next year, and here are some thoughts on my mind:

* If a kid is getting a high 70s grade currently in my honors class, would that kid benefit from dropping down to regular math and getting more in-class reinforcement / practice? (We can only do so much practice of the same thing in honors class before all the other kids get bored.) What if that kid enjoys the pacing and rigor of an honors class, but just has trouble mastering everything at that pace?

* If a kid is consistently acing my regular math class, but isn't much of an adventure-seeker in math, would they be suitable for honors?

* If a few kids who are pretty sharp in honors geometry wish to take Algebra 2 over the summer so that they can "move along" on the track and end up taking Calc BC in their senior year, is that a bad idea (or is that just my old-school opinion that no one should be squeezing a year's worth of algebra foundation into 6 weeks)?

* If our policy is that any senior who fails a course won't graduate on time, how can I encourage my struggling juniors to take non-AP Calculus next year to improve their college-readiness?

Thoughts or recommendations? I worry about the best placement for every kid, because long after I am gone from this school I would still want them to be properly challenged and to be able to enjoy math at the same time.

-------------

Addendum: The kids are coming around to agreeing that they shouldn't be squeezing Alg2 into a 6-week summer course. It helped that I said, "This other math teacher who graduated from MIT and teaches at a very prestigious private school in the States thinks it's a bad idea," and it also helped that my former Alg2H kids (now they're in Precalc honors) walked in on the conversation with one kid, and they collectively shook their heads to say that it's a BAAAAAD idea and that everything you see in Alg2, you will need in Precalc. Also, one of my current Geometry Honors kids told his friends, "Remember how so-and-so took Geometry over the summer to skip ahead? I gave him an easy -- facilisimo! -- Geometry problem the other day, and he couldn't do it. Taking math in summer school's a bad idea!"

## Monday, March 14, 2011

### On Raising the Bar for Tenure

I feel very emotionally affected by the mess that is the US* politics of education. I hate the thought that I could be forced to stay in international education for years because my job in the States would be too unstable once I re-enter the public education sector. I got into the business to help kids, and although there are kids who need help everywhere in the world, the places where I can make the most direct impact are in inner-city schools where I could potentially really change some of their lives. And I would like to think that at some point in the future, regardless of where I've been or what I've learned, I would return to do my part. And I would also like to think that I would be able to find an urban school with like-minded teachers and administrators -- one where I could potentially stay happily for a long, long time.

And God knows that when I do find myself in such a situation, I wouldn't want to be additionally dealing with the uncertainty of whether I would be able to keep my job the next year. In enforcing unpopular policies, the States* is definitely not encouraging teachers like me who have the option to work elsewhere, to return to the States. (And I'm only a young-ish 5th-year teacher; think about those who are much more qualified/experienced than me, but who have a family to raise and therefore have much more to lose. Would you return to the States right now with your family amid such a mess, in hopes of saving the world, one child at a time?)

All of that is a preamble to this: I really resonated with this blog post, which I think offers some great suggestions for balancing how much we protect young teachers versus how much we protect the more experienced teachers. If we raise the bar for receiving tenure, then we would less likely have to get rid of qualified teachers from one school simply because we need to find places for other tenured teachers. It would give everyone hope for working towards that level of job security, by actually doing a good job.

Am I just being naive and hopeful that there could actually be a solution?

*Obviously, the policies vary per state, but it seems like it's the same problem all over -- not enough union or too much union. Either way, as a young teacher but also someone who intends on sticking around in a system for the long haul, I would be screwed either one way or the other.

## Saturday, March 12, 2011

### Learning from My Mistakes

Both last year and this year, I have taught various methods to finding the circumcenter of a triangle. The reasoning is this:

1. It's interesting. Circumcenters are equidistant from original vertices A, B, C, so they give rise to certain problems such as "where is the best location for placing a new hospital/communication tower?"

2. The geometry of circumcenters is beautiful; it can be found by folding each point on top of each other point* (thereby creating 3 perpendicular bisectors kinesthetically), and afterwards its location can be verified by drawing a circumscribed circle that goes through all original points. This also helps to reinforce the circular property of equidistance.

*And this year, since my kids were the ones that came up with the "folding" algorithm, it makes total sense to them why that line is the locus of equidistant points from vertices, and why the circumcenter must lie on the point of concurrency of all those perpendicular bisectors.

3. The algebra of perpendicular bisectors is a nice way to loop back to line equations, midpoints, and perpendicular slopes. For the more advanced geometry kids, the algebra of finding a non-integer coordinate circumcenter also brings back systems of equations algebra, since that is the point (x, y) that needs to satisfy all perpendicular bisector equations.

But, last year my students had a lot of trouble with the algebra part. This year, we're done with the kinesthetic part and they don't seem to have any trouble with the overall concept. We started looking at the algebra, and originally I started it the same way I did last year -- by running them through the list of properties that a perpendicular bisector should have, and using those properties to help us write the equation -- which sounds good in theory. These regular Geometry kids can follow conceptually what I'm saying (since they have a strong conceptual understanding of circumcenters through our various activities and demos), but then when I let them follow up on the algebra example by doing one of their own, all hell broke loose.

Naturally, I thought in my head: I need to back the heck up!!

So, I made the following worksheet for them yesterday (ouch, algebra on a Friday!), and it went really well. The worksheet had different problems of varying difficulty; initially I would give them either the original slope or the midpoint already found (or both), and expect them to find the missing pieces and to find / graph the perpendicular bisector equation. (They graphed to check visually whether their perpendicular bisector equation "looked right" relative to the original segment; I refused to tell them whether or not their equations were correct on the first page.) Then, the worksheet scaffold up to them doing the whole process of perpendicular bisectors by themselves, and finally to finding the circumcenters using the intersections of those graphed perpendicular bisectors.

I found that by breaking it up like this into little pieces, it finally started to make sense to kids and they were able to "see" how the perpendicular bisectors, once they had finally understood how to find them, would lead them graphically to the circumcenters. (To be sure, these regular kids were having a lot of basic algebra issues. This exercise also helps them to zoom in on those, because it makes it relatively easy to figure out which part your mistake must be coming from, if half of the problem was already done for you. Because I'm trying to be less helpful and to force their independence, I also told them they needed to find their own algebra mistakes and not rely on me. Only very seldom would I help a kid diagnose that their midpoint wasn't correct, for example, by asking the kid to find that point on the graph and telling me whether that location appeared to be the correct midpoint.)

--Score! ...You know, it's funny. These are the traps I should have been able to avoid even as a first-time Geometry teacher last year. You can't teach kids a whole algebra process at once, even IF they already have the conceptual foundation. They need to already have a solid understanding of the individual algebra pieces, before they can start to put the whole process together end-to-end. It's something that I always forget the first time I try to teach something. But for some reason, it never occurred to me last year that this was what I was doing wrong. More practice doesn't automatically lead to more understanding; better and more thoughtful practice leads to more understanding!!

## Friday, March 11, 2011

### A Soft Approach to Holding Kids Accountable for Learning from their Tests

I know that a lot of you do the SBG thing so you don't believe in test corrections. Well, I have to admit that I've never been sold on it one way or the other. I've always thought that my job as a teacher is to create as many opportunities as possible for apathetic kids to learn. That means creating insanely accessible and stimulating and (as much as possible) not boring lessons. It also means giving them homework and checking it off for completeness, in order to encourage kids to stay on top of each topic. That also means consistently giving kids practice quizzes and tests with numeric solutions and time provided in class for questions / feedback, so that the material on the upcoming assessment seems concrete and doable for every kid. It also means allowing them to do corrections on quizzes and tests for partial credit, so that they'd go back over their own mistakes. And it means making videos and uploading them online, even if I'm not sure how many kids are going to be using them.

But, all this, for what?? NOT because I'm trying to reward kids for some sort of "nice" behavior in some sort of points game called school. I do it because the points don't matter to me; I don't care if kids "win" and come out of the other end averaging 70s or 80s or 90s in my class, as long as it means that I've successfully tricked them into doing hard work for me the entire time. I don't care if they got some buffer points for homework, if that means that their overall understanding went up and we can get to some trickier material during the unit. I do those things because I'll do whatever it takes for them to be thinking hard daily in my class. I do it because I want to create as many opportunities as possible for them to learn; it's like I'm leading the horse to the water. Quiz and test corrections, for example -- why are you against assigning points to them? Is it because you think that when kids correct their tests at home, it doesn't show real mastery? For me, I allow them to do corrections because I want to give them an extra reason to be sitting down with their quiz or test, and that incentive had better beat out all of the social network and Blackberry temptations. If they're irresponsible enough to copy off of their friend's quiz or test answers without even bothering to look and try to somewhat understand the answer -- well, that lack of effort will come back to bite them later on in my class, and in life. You can be sure of that. (I think of it as a karma of intellect. I help enforce the karma of intellect by making sure that every week in a unit, the level of difficulty of the material is ramping UP in my class. If a kid is happy with a 60% on a couple of quizzes and doesn't try very hard to look them over, he or she'll definitely get a 40% or lower on the test. Almost guaranteed.)

But, anyway, all of that said, I admit that quiz and test corrections are not bullet-proof. They're just a way to encourage all kids to remediate the material without risking lowering their grade further. (Most of my truly struggling kids are NOT confident enough to show up for a re-test, even if they might be able to do better the second time around.)

This week though, I think I've found the perfect complement to this test corrections thing. The kid has to come and explain the work to me, line by line. I swear, I did that with my 11th-graders (2/3 of whom had failed the last test, remember?), and each one of them rattled on beautifully about math for 15 minutes while walking me through their self-corrected 2-day-long chapter test. I would periodically stop each of them to ask questions, and for the most part I was extremely satisfied with their improved understanding. One kid said after I accepted his corrections, "I learned a lot from this!"

So, I don't care that the kid got those points through corrections. The worth is in them sitting down with me and explaining every problem. Even if they might not remember all of this material a month from now, they'll remember the sense of confidence that came with their few days of hard work in correcting this exam, and that'll do good to their relationship with math.

So, I'd say that my chat with my juniors is working. In fact, I've never been prouder of them since August! :)

## Thursday, March 10, 2011

### Beyond the Algebra of Composition Functions

Now that my kids are comfortable with writing composition function equations, I gave them a few word problems to illustrate:
1. why composition functions can combine functions that have different types of domains. (For example, function f takes in dollars as domain, and function g takes in time as domain. It's possible to get an equation that represents f(g(x)). See first problem in Part 2 of the worksheet.)

2. how composition functions combine step-wise dependencies to represent them all in one swoop!

(The examples are a bit silly. But, they are intuitive and easy enough for kids to grasp/follow. After this, I made them do a bit more heavy-duty problems in the textbook, that are less light-hearted and a bit more "real", but also less fun.)

Check them out! (Part 1 is adopted from a lesson from NCTM. I just re-formatted the questions and re-worded them quickly. I was a bit scared by how long it took my 11th-graders to get through that first exercise.) I think the worksheets were pretty effective. I'm sure if you did it with a more accelerated class, they'd breeze right through this, and it'd help solidify their conceptual view of compositions.

--------------------------

I also have an idea for how to teach inverse functions this year using a sort of telephone game. I haven't tried it out yet (that's for tomorrow's class), but I'm thinking of starting the class with writing a table of first / last names on the board and asking kids to evaluate
f(Isabella) or f(Alvaro) for last names. Then, I'll introduce / review that the reverse lookup notation is
f-1(Alvarez) or f-1(Garcia) and that this is called the inverse function.

Then, I'll call a few kids to sit in front of class in a row, and they each will get to pick a secret basic operation +, -, x, with an operand. For example, "+ 3" or "- 8" could be what they secretly choose. I'll give the person on one end of the row a number, and he or she would do the operation in their head, and then tell the result to the next person. So on, until they get to the end. Say that the original number is 10 and the final number is 61, I'd write f(10) = 61 on the board. Then, I'll ask them to go backwards, starting with the last person with the number 61. Each step, they should "un-do" their own step by doing the opposite of what they did before. That way, by the time they get to the front again, we should see that f-1(61) = 10 happened by reversing each operation AND reversing the order of each operation. And I'll use that to introduce how to write equations of inverse functions!

Example: First person secretly chooses "add 3", second person "times 2", third person "minus 9", fourth person chooses to square.
f(x) = (2(x + 3) - 9)^2

That means that on the way back, in order to use the output to find the original input, we would need to: First square root, then "add 9", then "divide by 2", then "subtract 3".
f-1(x) = (sqrt(x) + 9)/2 - 3

I think the nice thing about this demo is that we can repeat this process quickly if kids have questions about any part of the classwork/homework. (I'd just choose kids to represent each step of the operation, and have them sit in a row to demonstrate how the operations reverse their nature as well as their order.)

...I'm excited about this!! I think it will work and make sense to the kids!!! (Of course, I'll also teach them the "short cut" of flipping x and y and solving for the other variable. But I think that comes later, once they have a foundation of what inverse operations are and why they work.)

The other nice thing about the chairs exercise is that we can use it down the road to illustrate why f-1(f(x)) = x, by putting twice as many chairs in a row, with the middle two operations canceling each other out, and then the next pair canceling each other out, etc. Example:

x --> Add 3, Times 4, Minus 1, Plus 1, Divide by 4, Subtract 3 --> you get x back, obviously!

Thoughts??

Addendum: The telephone game worked fabulously! It also was a good anchor for me to come back to in order to explain to kids why inverse function has nothing to do with flipping the signs. All I had to do was say to kids, "When I gave them the reverse input while going backwards, I didn't flip its sign, did I?"

Using the first and last names as warmup also had the added benefit of a giggle factor, when I explained to the class that f(Cuellar) = undefined. (Cuellar is the last name of a very silly and likeable kid in the class, but since it's the wrong type of domain value, it cannot be evaluated. The class thought it was funny that his evaluation gives an error.)

## Tuesday, March 8, 2011

### Happiness!

Four very happy bits of math teaching news:

1. I have launched my new "Precalculus algebra skills" videos website! Check it out. Some of my students (not sure how many... mostly the strugglers, I guess...) are very excited about this extra resource. Or at least that's what they say. We'll see if they actually follow through and watch the videos. The videos are very detailed and so are longer than the 2- to 3-minute length that was recommended to me by a reader. That might change in the future, but I'm not sure yet. (It really depends on student feedback, I guess.)

For now, the videos are just there on a voluntary-access basis. That might change in the future as well, if I can be sure that every kid has access to them from home. (ie. 6 weeks before the final, I might start assigning 2 or 3 videos a week, or something, as spiraling review at home.)

2. I'm going to Park City Math Institute in July! woohoo! I'm super excited. (It's a summer program for math teachers and other math geeks. I've read really great things about it!) It's going to be really interesting / probably very hectic, because I'll be bringing my same 2 suitcases-worth of stuff that I'll be lugging to Berlin. Berlin is expecting me to report immediately after the PCMI summer program so that I can begin working on my immigration paperwork -- I actually had to negotiate a later starting date with their HR department, in order to make this happen. Which means that I'll have to completely move out of El Salvador before I head off to the program. Which means that things can get verrrry interesting in June. :)

3. I came up with a GREAT new way to teach bisectors as a locus of points satisfying a property!! I gave my regular Geometry kids a Do Now where they had to:
A.) Copy down the definition of a locus (with examples: a circle is a locus of points equidistant from a center, and a line is locus of points (x,y) that fit into a certain equation y=mx+b).
B.) Draw two points, M and N, and find 5 other points that are each equidistant from both M and N.
C.) Draw an angle YXZ. Then, find 5 points that are equidistant from ray XY and ray XZ.

After the kids tried parts B and C for a good few minutes (and most of them had figured at least part B out), I picked a volunteer to stand between a plant (representing M) and a chair (representing N), so that he/she's equidistant from both objects. Asked the kid to step forward away from both objects, but still remaining equidistant to both the tree (M) and the chair (N) at all times. After the kid took a few steps, we noted on the board how the kid had no choice but to walk perpendicularly each step away from the original segment MN.

We then repeated the same exercise but using the corner of the class as the original angle. I picked a different kid to start in the corner, to walk away from the corner, but always staying equidistant to both walls. We noted that they bisected the angle, and connected this to the Do Now problem.

After these demos, kids thought this concept was so straight forward! The idea that perpendicular bisectors (or angle bisectors) contained an infinite number of points equidistant to both endpoints (or rays) became obvious to them. We drew the diagrams on the board, connected the points using a line, and discussed why that line is the locus of all points that satisfy those equidistant requirements. I was really excited because I believe that this is a really abstract concept, but making kids act out the points really made a huge difference!!

We also used wax paper to explore how to find a perpendicular bisector by folding. I gave these regular Geometry kids pieces of wax paper, told them to draw points A and B, and told them to figure out how to get the perpendicular bisector by folding. Everyone figured out that A has to go on top of B. Silly me for actually teaching it to them last year!! (Apparently it's just common sense.)

Then we went into this pizzeria / bisector project from NCTM. It has a lot of good math in it! (I've used it once before; thought then that it was really great as well. I use the scenarios where there are 2 pizzerias in town, versus 3 pizzerias in town, versus 5 pizzerias in town. Each one really highlights different skills, and do not become monotonous for kids in that succession. The only very time-consuming part is that in the scenario with 3 pizzerias, you have to find the area of each region by counting the blocks and estimating when the fractional blocks make a whole block. For 2 pizzerias, you can still use a trapezoid area formula, so it's not so bad.)

On Day 2 of the pizzeria project, we illustrated the definition of circumcenter again using objects and people. This time, I had two volunteers, one starting off in the middle of a plant and chair #1, the other starting off in the middle of a plant and chair #2. As they each walked forward along the perpendicular bisectors, the class observed how at some point they collide. And I stopped them and said that that point of collision is called the circumcenter, and at that point they are equidistant to the plant and BOTH chairs! Later during the pizzeria project, when kids needed to explain the significance of a house that was located on all 3 perpendicular bisectors, they immediately recalled it being the circumcenter and recalled that it was equidistant from all 3 pizzerias! Brilliant!!

4. Even though I had been a bit skeptical of my own group activity of writing composition formulas and analyzing domains using a "playing deck" of function cards (see bottom of this post), my 11th-graders LOVED it!! And every child's understanding of composition functions improved visibly between Round #1 and Round #5. By Round #5, they were consistently getting the equations and the domains correct. I was SUPER happy!!!

Love my job. Love, LOVE!

## Monday, March 7, 2011

### Making my First Math Video

I've been thinking about the inverted classroom model. There seems to be a few key advantages like getting kids to be more proactive about their own learning and allowing them the ability to rewind or re-watch as many times as it takes for them to absorb all of the important info, but I don't like it as a way to introduce skills and topics. The non-interactivity of monologuing about math for 10 minutes cannot be very effective in activating prior knowledge. Even when I do a mini-lesson in class, I like to call on kids to provide feedback, so that I can make sure that I am pacing the class appropriately for the slower learners of the batch and that I'm clarifying and re-wording parts that may seem unclear to them in my initial description. The other issue is that my kids don't really have trouble recalling simple facts or concepts. It's when they all mingle together that my kids start to get confused. (And although we work out their issues in class, they don't always remember what we did in those n-step problems a few weeks/months later. --Shocking, I know!)

But, I am considering implementing something like this to help kids review the more difficult parts of a past topic or assignment that has already been introduced / worked on in class. That way, they can look through the archive and only tune in to the "episodes" that are giving them trouble. Also, this means that if they do not remember how to do something that's a bit complicated, they can always go back and find a relevant problem and watch the video to see how it was done. (Versus my current model, which is we would discuss their difficulties in class, and who knows how good their annotations are going to be in helping them work out the entire process later on??)

In fact, I am going to begin making a video archive of problems soon. (Why wait until next year to try this out?? March seems as good a time as any.) My one class where kids really need extra review is in Precalculus, and it seems like we're always running out of time to teach new material, that we can only do so much review in class for the material we have already learned/practiced. I am going to experiment with making some videos about current and past topics/problems, and dumping them onto a webpage so that these kids can look them up as they need to instead of always seeking me to re-explain the same things.

Addendum: So, since I'm one of those impulsive types, I abandoned this blog entry midway through and went ahead and tried to make a video last night! The video wasn't half-bad in quality actually, even though holding the camera in my left hand and writing while standing (easier to hold the camera steadily that way, while leaning my wrist on a box) and thinking about what to say at the same time (and trying not to mess up) was quite tricky. In fact, there was a noticeable pause in the video when I had to do a simple subtraction, because I was just so distracted by everything that was happening at once. I also got cut off half-way when the camera ran out of battery, so I had to finish it off in a separate video -- fortunately, it was in a natural break of the topic. But in the meanwhile, if you can take a peek and give me some feedback on the math explanation and/or the format, that would be great! (I have to still work out the technical aspects of the job. Right now, after conversion into Windows Media Player format it is still too big for my taste and makes uploading kind of a nightmare. I tried converting into FLV and it got really grainy and difficult to see the letters on the page. Suggestions??)

Here are the links: Part 1 and Part 2 of how to find domains for combined functions. Are you able to see them on a regular internet connection? (My hope is that they're more or less stream-able.)

## Sunday, March 6, 2011

### On Open-Ended Non-Questions

I came across this wonderfulness today in my Google Reader, and it made me pretty excited! (I know, I'm a total geek.) I love the idea that you'd give the kids an open-ended situation and leave it up to them to show as much understanding as they can about the situation. Seems like something that works extremely well in physics -- a subject whose goal is to encourage kids to gain a multi-layered understanding of everyday situations using a combination of various "laws" and models. It's also really exciting to me because this process highlights (to the teacher as well as the kids) that conceptual understanding of the same situation will continue to build, as you accumulate deeper knowledge on how to drill down further into pertinent details. For example, take a situation where a kid is told that a hot air balloon flies through the air. At the beginning of the year, they might only be able to say that it's because the air inside the balloon is less dense than the surrounding air. A week later, they might be able to say that this is because the air is warm and warmer air is less dense. Another week later, they might be able to say that the warm air is less dense because the molecules have more kinetic energy and therefore create pressure and expand the volume of the balloon. Another week later they might be able to explain the heating mechanism and why hot air doesn't escape through the hole of the balloon. And even later, they might be able to predict what temperature the balloon would have to have in order to carry a certain amount of weight. (I'm not actually totally sure if they can do those calculations AND I'm pretty sure you'd teach those concepts at a faster pace. I'm just using those steps to roughly illustrate how an idea about the same situation progresses over time.)

It sounds like something I would like to try in my own classes, but two difficulties immediately come to mind:

1. Is it necessarily applicable to a math classroom? Although open-ended explorations in math are indeed possible and interesting, a vast majority of the math processes we practice in our classroom encourage kids to eliminate cluttering information that they don't need. Whereas physics encourages the learner to take something very simple and expand it into something complex and multi-layered and to consider all factors involved, math (in my mind) is more like a funnel that gets rid of all of the cluttering details and focuses in on only the most important details. Even much of the WCYDWT stuff is inherently begging a certain question to be asked, and then more or less inherently requiring some specific math strategy (with some real-world messiness, of course).

Or am I missing something here and my view of the goals of math teaching is really much too narrow? (I'm thinking out loud here. Feel free to jump in. My mind really isn't made up about this one way or the other. After all, it's clear that the early mathematicians never limited themselves to thinking about the fastest way to get from point A to point B.)

Anyway, personally, I think there is always value in "playing around" aimlessly with a problem, even if it doesn't immediately lead you to something productive. Sometimes questions arise that way and other times solutions arise when you least expect them to, just because you've meandered your way through most of the issues. Past a certain age, all the problems that are worth solving are not solvable in your head anyway, so some playing around is usually necessary and giving kids these non-questions encourages that line of open-ended thinking.

2. Let's assume that my first point was moot and that this technique is entirely applicable to the math classroom. The beauty of this process, as I have described above, is that you can give the kid the same situation at the beginning of the year, the middle of the year, and the end of the year, and their understanding should continue to build upon itself and to encompass all existing knowledge, plus brand-new knowledge. What types of problems would I be able to give that tie together all of the things they would learn in the course of a year, so that they could demonstrate understanding at all different levels? (Here I feel that physics has another natural advantage -- a bunch of the ideas you learn in physics are all interrelated.)

I have no easy answers. I'm going to have to think about this one, so that I could possibly implement something like it next year.

-------------

PS. Speaking of good test problems, I recently put a question on my regular Geometry exam that nicely incorporated some old and new concepts all in one shot. The question gave the kids 3 different y=mx+b equations to graph, and asked them to find the angles inside the triangle that is formed from these 3 lines. In order to do that, they needed to know: 1. how to graph lines (old knowledge), 2. how to use pythagorean thorem/distance formula to find edge lengths of the triangle (old knowledge), 3. how to use inverse trig to find the angles within the triangle (new knowledge). Synthesis!

## Friday, March 4, 2011

### Rollercoaster

It's been a pretty emotional week for me in Precalculus. It's hard for me to explain, but I reached a breaking point this week when I passed back a test where 2/3 of the class had failed. They had failed even though both they and I had been working steadily on the material for a while now. There was A LOT of material; make no mistake. They had a test on everything we had learned from early January to the end of February, and it took them 2 days in class to complete. The test included at least 2 major types of word problems (2-d and 3-d optimizations and piecewise functions), and various not-easy algebra skills, such as finding domains of function equations, completing the square, and function transformations. (I gave them one half of the test the day before a long weekend, and the other half of it the day after.) But we had also spent 3 days reviewing, and I had given them practice problems very similar to those on the test, just to jog their minds along the lines of my expectations. A bunch of the problems they had already seen very similar stuff to, on the previous 3 or so quizzes.

So, I reached a point where I had to have a serious chat with the class. I told them that I thought that their grades were, in short, disappointing. I said that it showed me that they didn't go back to make sure that they understood the old quiz problems, and are just more or less copying quiz corrections without really making sure that they have understood the material. I also said that if they had really tried to understand the material after each quiz, they would have a much easier time than cramming 3 days before the test. I told them that from now on, I wasn't going to accept any test or quiz corrections from them without them verbally walking me through every step. If they couldn't explain something, I would send them away and tell them to wait a few days before coming back to me. One girl spoke up and said that there were a lot of topics on the exam, and it was difficult to study. But this girl in particular had done well on Day 1 (not having known what was going to be on the exam for that day), and very poorly on Day 2, after the weekend. So I didn't hesitate to point out that she wasn't being truthful about why she had failed the exam.

Besides that, there weren't really complaints; kids knew that they weren't doing everything they could to be where they needed to be. I was UPSET. I told the kids that it really bothers me to think that we are at the cusp of going into brand new math topics, and they're struggling so much with things that are more or less review of previous years' material. I also told them that I know that a lot of them think that this is the last year they "have" to take math, but that it would be extremely unwise for them to not take math next year, and to expect to go into college having skipped a year of math. I said, "You guys can barely remember what I taught you two months ago! How are you supposed to remember it two years from now??" The class was pensive, and quiet. I also told them that I know that they need to study more at home, because in class EVERY SINGLE ONE OF THEM can always do the work with some help, which means that the material is NOT out of reach and my explanations are not too obscure for them. The problem is that they only think about math during the 4 hours a week when I see them. I said, "Math is a lot like playing sports. If you want to get good at something under pressure, you need to practice the same thing more than a couple of times. You wouldn't want to go into a soccer game having only practiced once a few days before; why would you do that for math?? I stay here after school everyday; what are YOU doing to improve your grade?? If you're not willing to stay with me after school everyday -- that's OK -- get a tutor! Some of my freshmen have tutors; there is nothing wrong with that. Only 1 of them is failing out of all 4 of my freshmen classes. Do what you need to do and make positive choices! Get it together!!"

Anyway, as usual, I thought to myself afterwards that I had perhaps been too hard on them. A major reason for their poor understanding can be attributed to their recent ridiculous absence patterns. After all, how can you be expected to do well on a two-day test when you've recently missed an entire week (or more) of school?? Another reason for them doing pretty badly, as one parent pointed out in a conference with me recently, is that they have poor basic skills coming into my class (for reasons I won't discuss). The parent was concerned because she recently started helping her kid with math, and she realized that her 11th-grade daughter can only calculate rise/run correctly sometimes. (And, even though I review these "old" skills whenever applicable in class, we have to be realistic here -- how many slope practice problems could/would I assign to a class of 11th-graders?) But, the result is that they often make silly algebra mistakes in word problems, even though they have the big picture of what they're supposed to do.

I have very complicated feelings about all of this. But, after my chat with the kids, they started miraculously showing up for extra help. Yesterday, 3 of them came after school. Today, 6 of them came after school. --AT 3PM, ON A FRIDAY?!?! I told them, "You guys sure choose some interesting times to be motivated." One of them who had NEVER come to see me before said to his friend, "This stuff is so easy now!" I rolled my eyes at him and said, "That's why YOU should have come to see me BEFORE the test."

I don't know what to think about all of this -- not about their poor state of affairs, nor about this sudden surge of motivation. I can only hope that their motivation is not just some sign of short-lived guilt. Because this class really, really needs that extra umph. None of my 11th-graders has issues processing complex material. They just need a LOT, LOT of work to get fluent at it. (And I hope that they're all planning on taking some type of math next year, so that all of this effort I am putting into them isn't going to waste.)

## Thursday, March 3, 2011

### Best Group Work Ever!

Continuing with my "This was a trig Do Now that I had used that worked well for me" thread, I used two Do Now exercises to illustrate to my Honors Geometry kids how to break down quadrilaterals without ever lecturing. First one looked something like this:
1. Draw an irregular quadrilateral ABCD with one right angle A; use the corner of your white paper to get an easy right angle.
2. Measure AB and AD only.
3. Measure all 4 angles of ABCD.
4. Draw a dashed line from B to D.

Then, I had the kids fold the quadrilateral back along the dashed lines, completely solve for everything inside triangle ABD (all angles and all sides), and then open it up and completely solve for all remaining angles / sides, and finally to find the quadrilateral area! (I had to remind them that earlier we had found the area of a scalene triangle to be A = (1/2)(a)(b)(sinC). Other than that they were uber-independent and got the whole thing quite brilliantly. The activity was again self-checking using protractors and rulers, so it helped to build their confidence in their own rather elaborate calculations.)

After this, we worked a little bit on Kristen's awesome trig project problems. Since my honors kids are sharp, I didn't need to elaborate further on how to divide up quadrilaterals besides emphasizing that we had seen how right triangles make up scalene triangles; now we were going to see how scalene triangles make up quadrilaterals as well, and to use that as our basis for analysis.

Day 1 of the project was a bit slow. I didn't need to help them too much, but they were working slowly in groups, trying still to grasp when to apply which formula. They were tentative about discussing the problems, and mostly worked individually. I saw this as a sign that no one was sure-footed enough to pipe up in a group, and it made me wonder whether my kids were really fully ready to venture into quadrilaterals on their own just yet. So, I decided that what they needed was another scaffolding activity, just to ease the transition a bit.

So, on Day 2, I devised another Do Now that furthered this idea of cutting quadrilaterals into smaller pieces and applying Laws of Sines / Cosine to analyze the individual pieces. This time, I wasn't so interested in them actually carrying out all of the calculations. I just wanted them to make a plan of attack, so that they can begin to see the pattern/big picture:
1. On a sheet of white paper, draw an irregular quadrilateral ABCD. This time, don't use right angles.
2. Measure all 4 sides of ABCD.
3. Measure only angle A.
4. Describe, step by step, how you would solve for all angles of ABCD.
5. Also describe how you would find the area of quadrilateral ABCD.

Seems innocuous enough, but the kids had to work hard for this one. Only a few kids were able to run with it completely on their own. Most figured out how to cut the quadrilateral into two appropriate triangles. After waiting a while, I started to give the class hints along the way, one hint every few minutes. (I picked kids from the class to share how to do the next step, every few minutes.) They struggled through this, but the process of thinking about it was very worthwhile. After this Do Now, the kids worked on the remainder of the project packet with much more fluidity and confidence.