Wednesday, October 24, 2012

Geometry Is Delicious!

We started our surface area and volume unit this week. I get excited about this unit because it connects so well to so many other topics and its easy to find quality problems. In the near future, for instance, we will be completing a numbers of Dan Meyer's 3Act Tasks because its just so easy to relate.
Anywhoo... we've been developing the formulas for the figures. We developed lateral area, surface area, and volume of prisms and cylinders yesterday. Today, however, I got inspired to move to spheres. How is the surface area and volume of a sphere calculated? Why?
I've never tried this before, but I thought I'd give it a shot. I brought in tangerines and clementines for everyone. Their first thought was, 'awesome free food.' We proceeded as follows: As the students peeled their fruit I said, 'Wait! Don't throw it away! Its really important!" Using string they approximated the circumference and calculated the radius. After a short lesson on how to properly use compasses, they drew circles of the same radius. 'How many circles do you think you can fill with your peel?' Estimates ensued, some big, some small. Just for fun, I asked them if they had a grapefruit instead, how many circles do they think they could fill. Surprisingly, I got a variety of answers. Some said they could fill the same amount, some thought more because it got bigger, some said less because it got bigger. Let's find out!
They proceeded to take the peel and fill in as many circles that they could.

Now, I've never tried this before. In years past, I was Johnny Boring and just handed them the equation. After some reflection as to how terrible those lessons were and how much better they could be, I collaborated with some colleagues and put this little activity together. And Boy Howdy did it go well! Seriously, it was awesome. Most students filled in four circles and put the connection of 4*pi*rsquared. It was great! And, bonus, my room smelled terrific! The students made connections, saw some cool stuff, all in all it was a great day and something I will continue to do in the future.
As an added extra, a former student of mine brought me in a baseball and I took it apart at the seams and used it to further illustrate surface area.
Now, to calculate volume of a sphere, I had them eat the fruit. That's it. We didn't derive the formula due to the fact that the derivation is awesome. And by awesome I mean gross.

To permanently (kind of) keep this lesson in my students' minds, I taped my example to a sheet of paper and hung it on my wall. I give it a week before the wonderful orange tangerine peel turns to slimy brown smelly mush. But when my students see that mush, you know what they will think about? Probably not surface area at first, but eventually they will. 

Thursday, October 11, 2012

Mumford and Math

I'm a big fan of Mumford and Sons. I only started listening to them last year, and quickly moved them to the top of my playlist. When their new cd came out a couple of weeks ago, I of course purchased it as soon as I could and it really is as good as everyone says it is. Two quality albums in a world where technology and catchy tunes are considered good music these days is uncommon. I began doing some research as to what makes them so good.
It turns out for their first album they sat and wrote each song, one by one, until it was perfect. Apparently it took a great deal of time and effort and produced some frustration at times. In the end they produced exactly what they wanted: an album with every song exactly as they envisioned it. For the new album they started to write songs in a similar fashion. However, this time around, they quickly realized how long it was taking to perfect every detail in each song before moving on. They changed their writing style for a bit at this point. They played a game: each member went to a separate corner of the house and wrote as many songs as they wanted, not worrying about the details, not worrying about the overall quality if it was 'album' worthy, not worrying about anything. They just sat, wrote tunes, and wrote lyrics. After the 10 minutes they each came together and shared what they had come up with, picking the ones that they thought were the best. Then, as a group, they perfected each song.
I've never heard of any group doing something like this, and it intrigued me. I got to thinking about Sir Ken Robinson and what he would say about this. Talk about creativity! Then I started to think how this could be implemented into my profession. It was clearly a good strategy to use in the music business when all of the members are experts in music. Would the results be the same for a group of expert teachers?
Now, I don't consider myself to be an expert teacher by any stretch of the meaning. I make mistakes on a regular basis, I teach poor lessons just like everyone else; I approach everything I do with an open mind thinking about how I can improve upon my job. But still, what would happen? I decided to put it to the test. During our department meeting today we gave this process a try. Here's how it went down:

1. We started by writing a couple topics down that fit into one of these categories: something I wish I could teach, something I have difficulty teaching, something I feel uncomfortable teaching. We came up with linear programming, simple harmonic motion, and law of sines/cosines.
2. I let each of my colleagues choose which one they wanted to work on (it ended up that half went to simple harmonic motion and the other half went to linear programming).
3. I asked them to write me a full lesson on their topic within 10 minutes, ignoring the details but simply by establishing a framework, opening, closing, etc. As a department we will fill in the details to create a high quality lesson.

I did this for a number of reasons. For starters, I was curious. What would be the outcome of this exercise? Would we really have some of the best lessons we've ever written or would they be just as good as what we've been doing for the past x-years. Secondly, I wanted to get my department communicating more about things over than 'do you have a worksheet/test for that?' I want us to be a social, collaborative group that feels comfortable bouncing ideas off of each other and willing to actually work together. Finally, I wanted to expand some people's comfort areas and get them to think outside the box. I'm not quite sure how each member plans their lessons, but I imagine its safe to bet they don't do it in 10 minutes. I'm also willing to bet that they don't do it with no materials in front of them, using purely what's in their head. I purposely did not tell them the details of this plan ahead of time because I did not want them bringing content maps, lesson plans, or textbooks to pull from. I wanted them to be creative. I wanted them to make the connections between the math and figure out a way to show it to their students. I wanted them to teach a concept in a way that made sense, not because a textbook said to do it a certain way. I feel that all too often we teach to what the textbook says, rather than teaching to the kids. This mindset bothers me and I wish to change it. For example, why aren't the trig functions, the unit circle, and the graphs all taught simultaneously? Sure, that's A LOT of information to through at the students at the same time. But doesn't it make more sense to show them how its all connected? Rather than most textbooks which have them in different chapters? You wouldn't have to go into huge amount of detail right away, but you could at least illustrate why it all works out the way it does. I wanted my department to take part in this exercise because I want all of us to be able to think about what is really going on in each of our subjects/topics. I want us to focus more on trying to develop good lessons and focus less on the material. By putting a time limit on the planning, I'm hoping that forces some creativity. By getting rid of the pressure of coming up with the details, I'm hoping that we can relax a bit and come up awesome introductions and summarizers and work together for the rest. I think that this can have a really positive impact if everyone approaches it in the right way. If anyone looks at this as a pointless exercise and doesn't venture out of their comfort zone, then it will turn into a quick way to produce the same mediocre lessons that have been teaching. I'm hoping for a positive experience and to try this again in the future, or at least to have members try it on their own when they get stuck. We shall see what happens.

Wednesday, October 10, 2012

Connections! That's what we're all about!

So, what makes math so awesome? A lot of things, in my opinion. One of those is how ALL of mathematics is connected. Geometry, Algebra, Pre-Calc, Stats - they're all interconnected in the web of awesomeness that allows every topic to make perfect sense. And yet I believe there are a lot of math teachers out there that don't explain or illustrate these connections. Why? Hopefully I'm wrong, maybe its just within my geographical area that it exists.
I remember my 'ah-ha' moment in high school (there were actually numerous moments that kept reinforcing each other). In pre-calculus, I remember my teacher showing me how to derive the quadratic formula. Wait, what? This isn't just a random formula that some old dude came up with that just-so-happens to work? *Mind Blown* Then again, later in the course, we started doing trig. identities. Wow! I think I'm the only student in the class that figured out the awesomeness behind trig. identities. Everything in pre-calc made sense after that point, because you really can derive anything. Wow!
Within my department we set a goal for the year to create lessons that require higher-level thinking/questioning from our students and allow them to dive deeper into in the material. I know when I started teaching, I was part of the 'I teach what's in the book, how its in the book' crowd. Now that I'm comfortable with the material, I've drastically changed my style. My goal is to teach the student, not the curriculum. Sure, I have a set up standards and topics that I have to teach, but guess what? I'm going to cater to the students. If they ask me a question that's related to what we're talking about, but would normally come up later in the semester, I'm teaching it now. Why should I turn down students' interest in math? They've had enough math teachers in their past destroy their possible love of the subject already; I want to rebuild that.
Anyway, all of that to say the following... In my geometry class yesterday we started the area/perimeter unit. Its obviously something that high school students have seen and mastered already, so I posted this problem to put a spin on it and give a little challenge:

Since we just finished working on some right triangle trig, most students were able to calculate the height and go from there. However, I had a student set it up incorrectly, but get the correct answer. Here's what she did:
Oooooooooo. She recognized her mistake in copying it down wrong, but was still curious if her still correct answer was a fluke or would it happen every time. Now, there are some teachers that would simply say "it was a coincidence" or "you set it up wrong so therefore you're wrong" or ignore it all together. However, I saw this as an opportunity to teach and expand minds. Why is the sine of 105 equal to the sine of 75? Long story short, we got into a discussion about the unit circle and a more detailed reasoning as to where these trig. values come from, and made the connection among the sine function and supplementary angles. The students were HOOKED! They learned some pre-calc in geometry! Holy crap, is that allowed? It's encouraged.
Throughout this team-teaching that I've been doing with my colleague, one of the areas we've been focusing on in making those connections among math concepts. We've both noticed that when we're relating topics to other topics to other topics to other topics and they're all from different math areas/subjects, the students are ridiculously engaged and focused. They Learn the material instead of ignoring it or memorizing it. Its awesome. Even students who don't really care about math can see value in where it comes from. There is something perplexing and fascinating about knowing where it all comes from and how it all works together. 
The 'why' in math needs to be focused on more in classrooms. Students appreciate the subject when they can make those connections and the subject has meaning.
So why don't more teachers do this? I'm not quite sure. Maybe its a lack of confidence in the material? Maybe its laziness? Maybe its because the textbook doesn't go in that order? Maybe its one of a hundred other excuses/reasons. No matter what, its something that needs to happen to increase our students' knowledge. 

Wednesday, October 3, 2012

Quick Question (well, maybe not quick)

How does one approach fellow educators whose view on education is this:
Rather than this:
Last night, Mr. Pershan tweeted something interesting: "What's a question about teaching that you wish you knew the answer to?" This got me thinking a little bit. And then this morning I thought more. My original response was "Is it possible to get all levels of learners to see, understand, and appreciate the interconnectedness of all math topics?" I tried to think of Mr. Pershan's question from my current standpoint, knowing what I know now as opposed to what questions I might've had my first year teaching. This morning I thought of the above question. Its something that I've struggled with ever since I made the transition myself. On a regular basis, across subject areas, I run into teachers that stuff facts and notes into students' brains and expect them to just regurgitate them. To me, that's not really the point of education. Sure, there are certain pieces of information that we need to learn and remember and use exactly as we learned them, but the majority of education should be learning how to problem solve and apply what we learn to new situations. I always have students tell me they're never going to fly on a plane that is landing at a 5 degree angle at an altitude of 3000 feet, so why do they need to know how much distance they need to land? I don't really have an argument for them; they are most likely correct. However, I don't teach them how to solve this problem exactly the way its written. I teach them the trig. functions and how they relate to the right triangle, and they THINK about how it can be applied to various scenarios. I hardly ever give the same type of problem more than once because all too often students use the same steps to solve everything. I'm trying to break them of this habit. I make it very clear to my students at the beginning of the year that I will teach them how to think because currently they don't know how; they only know how to read directions. After a few weeks, they start to get it and they start learn how to apply knowledge to all kinds of situations. They think outside the box rather than look into the box for steps 1, 2, and 3. Its an amazing transformation to witness, and the growth I see from the beginning to the end of the semester is quite awesome.
Back to my original question, how do I transfer this mindset to teachers that don't share it (or how do I make teachers realize this isn't what they're doing when they think they are)? Maybe that's not my job (but when I want what is best for students, its hard to ignore these things). After all, I'm not claiming to be 'Johnny Know-It-All' on this topic. I can do it with my students mainly because I've been teaching the same course for 6 years and have been able to perfect my lessons to reflect this process. I'm sure if I taught a new course next year, it would take some time to develop lessons that develop this mindset in students. I certainly don't have all of the answers, but I feel that I'm able to identify what I consider good teaching from bad teaching (and I have more than 2 years experience). In this 'team teaching' that I'm doing with my colleague, he is showing me all kinds of stuff that I never thought about in his lessons (there are times I feel I'm learning as much as the students). I've always been open to criticism, and that's why I've been able to change my teaching style. I was lucky enough to have a fellow colleague of mine take me under his wing and help me get my students to where they need to be. Combine that with the math twittersphere and blogosphere and its been a super fun process; doing all kinds of research and applying them to my teaching has been more beneficial than parts of my college education.
Maybe that's the first step: recognizing that there needs to be a change. I'm open to it, and I know others that are too. But what about those that aren't?

Note: the first image was taken from Mr. Pershan's parent night notes (@mpershan)

Monday, October 1, 2012

Go Team! Teaching

It was a cool Monday morning; students were trudging through the building as if they were just rolling out of bed and trying not to fall down the stairs. I was huddled at my desk trying my best to prepare for the day and change the lives of my students. The sound of squeaking shoes and muddled conversations was beginning to break my concentration. I tried to push through, but soon realized that it was no use; time for my morning duty. As I stood in front of my classroom door, I could now put faces to the growling sounds coming from the hallway. I did my best to transfer my energy for the day to my past and current students. I even transferred a 'Good Morning!' to those that I did not recognize, but alas, no results. As time ticked away, a fellow colleague of mine stopped by my room to ask a favor. He had a meeting during his first block class and asked if I could cover for him for a few minutes. "No problem," I said, "what are you teaching today?" He responded with, "The students are learning about function operations. I'll leave options on my desk for you if you don't feel comfortable teaching a lesson." He handed me his notes for me to look over if I was interested in teaching, and I thought to myself "Sure, I can do this on the spot." During homeroom, I read over his notes, quickly trying to come up with something interesting and engaging that I could present to his students. I had no interest in 'out-teaching' him in any way, I simply wanted to put together a quality lesson. I certainly did not want to take away from his class in any way. Fast forward 15 minutes and I'm standing in his class getting ready to teach. I was given the option to allow them to continue their classwork, go over their homework, or teach. I was feeling confident in teaching, and I recognized half of the students from previous years, so I went for it. I began teaching function operations and function composition to a pre-calculus class that I had never planned for before and it went Xtremely well. While there wasn't much exciting to the lesson, I was able to build off of their prior knowledge and construct new ideas that stuck. However, the best part was yet to come. About 10 minutes into my teaching, my colleague returned to the room. Since I was in the middle of a thought, I continued until I came to a stopping point. I asked if he wanted to take over and he sat down and let me continue. No problem; I was having a grand ol' time. It was a unique experience teaching a lesson and having the students participate and listen 100%. I'm used to freshmen that are still all over the place and seniors that have a hard time understanding the value and beauty of mathematics. These juniors were on top of their game and begging for more information. It didn't take my colleague long to notice this (he experiences it on a daily basis with them) so after a few minutes he jumped right in, helping me teach. Before I knew it, we were both teaching his students, and they were hanging on every word. We were able to bounce concepts off of each other, fill in gaps where one (me) might have missed something, and fully illustrate what function operations really represented. As the students led me through examples, he was standing at the smartboard illustrating what was going on through graphs. I noticed the students' heads constantly moving back and forth, but not in an overwhelming way. They were truly engaged, hanging on our every word. Jokes were told, knowledge was passed, lives were changed. I
In talking with my colleague the next day, he said his students remembered everything we had talked about in the previous lesson. Team teaching was effective, and super-freakin'-fun! He said the students loved it and requested that it happen again on a regular basis. I walked in later in the week and I could see the excitement on their faces grow, not because I'm awesome, but because they were hoping to have some more fun in math class! Fun in math class? Really? Is that possible?
I've always heard of team teaching and thought it was something that was only done in the past. But now I know why it was done, because it works! The lesson we did together was completely spontaneous and I think it was one of the most effective lessons I've ever been a part of. Just think of what we could've done had we planned it out ahead of time!?! So, obviously, we've decided to make this a regular part of our lives. Because of the positive outcome, we're going to get together and plan lessons and be guests in each others' room at least once a week. But doesn't that cut into your planning period? Yes. Doesn't that make you so mad? No. Aren't you going to ask for extra payment since your teaching more? No. This is fun, and its great for the kids. A better question is, why wouldn't we do it??
As we do this more, I'll try to get some pictures and post updates as to how it is going. I believe this week students will see more than one teacher in their room, but not for intimidation purposes...