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! :)

Miscellaneous El Salvador Updates

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!

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.

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.

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.

Are Your Kids Using Google?

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

A dad took his young son to the street and said, "Son, I am going to give you a task. You have to do everything that you can in order to get it done. OK?" The son nodded. The dad then pointed at a giant block of stone in the middle of the road and said, "Move that out of the way." The son went over and tried pushing the rock with all his might, but the rock wouldn't budge. He tried until sweat broke out on his forehead, but to no avail. Finally, he turned to his dad and said, "I can't do it, Dad. It's not possible!" And his dad replied, "Why didn't you ask for my help? Until you have asked for help from others, you haven't tried everything within your power to get it done." Together, they were able to move the rock to the side of the road.

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!!