Friday, November 30, 2012

Thoughts On Student Directed Curriculum?

WARNING: Lengthy, lack of visuals/humor. I'm going to hit you with some knowledge here (and I hope you, in turn, smack me with some feedback as well).

This is my sixth year teaching geometry and I'm pretty much at that point where I can walk in to my classes ask them what we talked about yesterday and I can go to town. I've always wanted to be at this point in my career, where I can spend time focusing on increasing the quality of questions and researching new ways of teaching rather than taking a tremendous amount of time just figuring out what I'm doing the next day, creating problems, etc. Because I have most of the ground work laid out, I've been able to try some really cool (and some not-so-cool) things over the last few years. For some of my units I've gotten rid of tests and created projects, for some I've made them more discovery-based and student-directed, and for some lessons I've created some really involved questions that require some incredible thinking from my students to solve. This year an idea I've been toying with is the student-directed curriculum. Its got some pros and cons to it and I'm not quite sure what I'm going to do.

Our geometry curriculum used to be like almost every other math class: follow the textbook. Unfortunately, we have UCSMP which is terrible in my opinion. When I rewrote the curriculum two years ago I had one goal in mind: organize it in a way that will make sense to the students and where connections can be made. I got tired to teaching topics and having to jump all over the place to struggle to draw the lines between everything. The beauty of geometry is that is all related and our students were not seeing that. Now, even though technically we jump all over the book (although no one in our dept uses the book anymore) I've noticed student achievement go up and we've been able to create deeper questions. The students don't care that they don't have a book to follow because the way that its organized makes sense to them. They would rather reference pg. 18 and pg. 262 in the same day than have everything disorganized and all over the place.

This semester I took the first three units (basic vocab, quadrilaterals, triangles) and mashed them together as opposed to teaching them separately. Now these units include things like all of the types of angles, symmetry, trig., and more, so they're pretty heavy on material, but I wanted to have it make even more sense for my students (if that's possible). As I thought about the curriculum, I realized that it could be organized in a number of different ways and still emphasize all of the interconnectedness of the world of geometry, but how could I have it be the absolute best for my classes? Or for anyone's classes for that matter?

 I started the year by asking my students what came to their minds when they thought about geometry. Their response: "Shapes and stuff." Me: "Name some shapes." Them: mur mur mumble mumble shapes blah blah ...and somehow I took it from there. I took the shapes they gave me and we started to dissect each one individually, exploring all the properties until there was nothing left to talk about, making connections as we went. I never had to say "Ok, we're done with that figure. Let's move onto the next one." because the properties naturally lead to more figures. Through student questioning I was able to completely cover the curriculum, plus some more.

Because this was my first time trying this, I limited myself to just the first three units as opposed to attacking the entire course this way. I'm teaching two sections of geometry and my hope was that I would be teaching them different concepts because their questioning would lead them to different places, but they'd end up at the same place in the end. I'll be honest, it got confusing. I no longer could stand in front of them and say "What did we do yesterday?" because I couldn't remember everything we'd discussed. I found myself keeping an unnaturally large number of post-it notes on my desk reminding me of what I've discussed with each class. I'd imagine when my third block asked my second block what to expect in class, they were surprised when they did something completely different.

I noticed that teaching this way caused achievement to increase compared to other years. Now, obviously I could just have an awesome batch of students (which I do), but I believe the way I taught had some impact as well. All of this has led me to ask myself, what if I taught the entire course this way?

Teaching an entire course through student questioning: innovative or something I should've been doing all along? Either way, there are some definite benefits and challenges to this approach. I believe my students would see some great success both in learning the basic knowledge and developing some higher order thinking skills. Teaching this way allows me to easily pull all kinds of topics together and potentially create some really cool projects. It would allow my students to really understand that math is all connected; its not separated into Alg 1, Alg 2, Geometry, Pre Calc, etc. but they're all based upon each other. It would also keep my students involved in the course. They would have complete ownership over the material because they would be determining what happens next. This also reinforces my philosophy of 'teach what makes sense, not what comes next in the book.' Looking through the PACCSS, I think I could hit everything with this style.

The tough part is that I have to almost be fully prepared to teach the entire course at any moment. Because I won't know where my students will lead me, I need to have everything ready to go on the first day. I'm sure I could predict a little bit as to where they would head, but I wouldn't know what they're going to do on a daily basis. If I taught this way I wouldn't want to push them in any direction unless they stall; I want them to be pushing me. Another challenge is keeping straight what I've covered and what I haven't in each class. If I would do this next semester, I teach three sections of geometry, that would be three different places in the curriculum simultaneously. I would be very fearful that I would forget to cover something or I'd start going over something in May that we discussed in February. While ideally this would be 100% student run, obviously there would have to be some questioning on my part to push them, which would give me some influence as to what's being discussed. My post-it note system of organization would fail rather quickly and I'd have to come up with something more efficient. I'd also need to create a new system for catching students up when they are absent. Maybe designate someone to constantly take picture of the board and post them online? There are some details to figure out.

As of now, I'm leaning towards doing this for one class instead of all three as a trial. This might help me get some of the details worked out before I push through entirely (of course, we all know what will happen if I do this: this will be the last year I teach geometry and I'll be back to the drawing board with new courses next year). It would also be fun to switch things up a little bit. I'm excited, and scared out of my mind, to try this. I'm not usually the most organized person in the world so this plan has potential to fall apart in a hurry. I'm hoping my desire outweighs any negatives that would potentially come out of this. The positives definitely outweigh the potential bumps in the road, and because of that I keep coming back to "I'd be an idiot not to do this!" I can't help but think of a quote from Mr. Pershan that I recently saw on Twitter (@mpershan), "If I'm not working really hard - if it isn't mentally exhausting, then I'm probably not getting better."

However, this style does kind of go against the 'common unit assessment' plan that my district has implemented for this year. Having common courses among teachers gets thrown out the window with this idea. Oh well... I gotta do what's best for the kids.

Do you, Mr. or Mrs. Reader, have any thoughts on this plan? Any positives or negatives that I didn't mention? Any ideas on how to overcome the negatives?

Thursday, November 15, 2012

What's the best way to clean up blown minds? With more awesomeness!

Similarity is a tricky topic; somehow it always ends up being more challenging to teach every year that I teach it. I think I learn something new every time that I hit this topic in geometry, which perplexes me because its a rather simple concept. Two figures, same shape, different sizes: done. Sure, when I start talking about the ratios between similar figures' areas/volumes, things start to get interesting, but overall, not a difficult idea to wrap my mind around. I feel like my students always have this approach as well. They find out we're going to be discussing similar figures and they get excited because we're going into an 'easy' topic, but by the end they always leave thinking, "Woah. That was intense." Perhaps I should stop telling them on the first day, "Hey guys good news. We're talking about similarity, one of the easiest topics in the course." Yeah, I probably set them up a little bit.
This year's enlightenment came unexpectedly when I placed a seemingly easy warm-up problem on the board yesterday. We've pretty much discussed similarity in as much detail as possible, so my students' brains are fairly swollen from the amount of knowledge they've gained up until this point (more on that in a second). The problem looked like this:

 "My classroom is about 30ft long and takes up about 700 sq ft. If I want to draw it to scale so that it has a length of 5in, how much space on my paper do I need to reserve for my classroom?"

No problem, right? Calculate your scale factor, square it, use that to calculate your new, smaller area. Done and done. I figured we'd be able to answer this question plus the three others I had on the board within 10 minutes tops. Right? Sounds reasonable? I mean, it is just a simple problem, nothing new. We've been talking about these relationships for the last two or three days. Welp, as it turns out, I'm bad at predicting these kind of things.
Here is brief visual as to what ensued (if only my whiteboard could've captured everything that was said. Unfortunately I had to erase some awesome ideas because I was running out of room):

(I apologize for my poor panoramic cut, slice, paste job)

This single problem generated more discussion than anything we've discussed all semester. My whole 10 minute idea went out the window, we spend ONE HOUR talking about this problem. Now, I'm not upset by this. I believe there are many teachers out there who at this point in the blog would begin writing about how annoyed they were that they had to reteach topics and what not. I had quite the opposite experience; this was one of the best hours of my teaching career! I had 10 different students show 10 different ways on how to solve this thing. It was ridonkulous (that's right, I went there).
Typically with warm up problems I #hashtag students to put their work up and explain to the class what they did and then we discuss as a class. For this problem, I had heard a good amount of students talking about it, so I decided that we would talk about it together rather than have someone put their solution up. I asked for some suggestions as to what to do and I had a student offer to calculate the width of my actual classroom, calculate the scale factor, find the similar width, and use the dimensions to calculate area of the scale drawing. No problem. Another student converted feet to inches first, calculated scale factor, squared it and used that to calculate new area. Awesome. Another student set up a proportion by first converting everything to inches, then calculating the width of the classroom and setting up 360/279.96 = 5/x. Cool. Another said (and this is where it started to get interesting) to set up a proportion like this: 700/30=x/5. This was a nice way to discuss how because we have area, the proportion should be set up as 700/30^2 = x/5^2 to keep the relationships the same. Great, some learning is going on. Yet another student said, "Mr. Brandt, I did differently. I think I did it wrong, but I still got the same answer." She set up 30ft/5in to get a scale factor of 6 and went from there. Another student set up 360in/5in to get a scale factor of 72 and continued on. Both worked. There were more solutions offered and all of those processes worked. Everything kept working out! I looked at those last two options and said, "Now wait a minute. We've got two different scale factors for the same problem, but they're both giving us the same answer." Discussion ensued as to why this works. I had a former math teacher from my IU observing yesterday and he started in on the discussion. My students still had some misunderstanding (and to be quite honest, at this point I had seen so many different solutions that my brain was slowly disintegrating) so they went and got another math teacher from across the hall and asked him his opinion. At one point, there were 27 students and three math teachers all working on this problem trying to come up with explanations as to why all of these different solutions worked. It was awesome!! I still had some skeptical students that weren't convinced all this was mathematically valid, so they had me come up with an alternate example so they could test each one out. I put another problem up and everyone got to work. I was pumped! This was education, this was school.
Now, I didn't go into as much detail as I should've for the sake of space and time (and your attention span) but these students really were working for an hour on this one problem. The entire time the entire class was discussing it, either with me or their neighbors. They wanted to bounce their ideas off of someone else to see if they were making sense. When I saw this whole process start to happen, I got nervous that I was going to lose some students. I wasn't sure if I should continue with all of these explanations or just say "Here's one solution, let's move on." But, instead, I put some faith in my students and let it ride. Best decision ever. We were discussing how to set up proportions and make sure that everything was in the right position, how to identify what proportion to set up, how to identify which relationship we've calculated, and the big question: do units matter when calculating similar figures (outside of ft, ft^2, ft^3)? It turns out, they don't (which was news to me). All throughout high school, I was programmed to always make sure I had consistent units, so I convert out of habit. My students didn't and they got the correct answer. Sa-weet!

There are some teachers that I can think of that would've taken this opportunity to say "Woah, woah, woah. We don't have time for this. We've got an answer and we've got other things to cover. Let's move on." And as I said earlier, this thought crossed my mind out of fear of what would happen, but not because of the curriculum. My students were learning; wait, I mean they WANTED TO LEARN. They were begging for this understanding. I believe had I said "Too bad kids" they would've revolted, they were that into it. Who am I to complain when my students want to learn geometry? I believe if I've inspired that kind of passion and desire for learning in my classroom than I've done my job. This is the end goal of education. My students got creative, solved the problem in many different ways, collaborated, discussed, debated, asked 'experts,' rinse, and repeat. 
This kind of problem solving and critical thinking needs to exist more in more classes. As I write this, I got an email from a student asking me how she can learn how to think because she can't do anything outside of the examples she sees in class. Students need to be taught how to be creative and apply their knowledge to all kinds of situations. The question of 'why' needs to be answered by teachers, and the relationships and connections to other topics need to be emphasized so students can generate that genuine curiosity and amazement. I know that when I was in high school, my performance skyrocketed when I started noticing how everything was related. The best teachers that I had included this information in their teaching and for the ones that didn't I asked for it. I see  too many teachers who are very by-the-book and if its not in the book than it can't be discussed. Books no longer run the curriculum. Teachers need to teach the students, not what's on a pre-made worksheet.

As my students walked out of my class, there was a mix of emotions: excited and overwhelmed, but in a good way. Minds exploded in my classroom yesterday, and I feel bad for the janitor. I'm hoping my room has since been sanitized so that I can do it all over again. 

Saturday, November 3, 2012

3 Acts = Desire To Learn

Last week we started working with 3D figures, surface area, volume, etc. We went through developing all of the necessary formulas, practiced using them, and the whole class was a set of all stars. So the natural thing to do was move apply these beautiful concepts. Initially I thought of starting this unit with application and developing everything from examples. Instead, since we just came off the area/perimeter unit, we used those skills to develop everything necessary for this. In my opinion, these two units are the easiest to find applications because everyone has to paint a room or put something together at some point in their lives. The natural progression was to throw some 3 Act problems at my students and watch what happened.
They started with Dan Meyer's world's largest coffee mug. The questions started flying, I provided information, and they were off to the races.

I like the image below. When I noticed the approach they were taking, it allowed me to generate some decent discussion among the students.
They took their mug and split it into 2D boxes, or so it seemed. In speaking with the group, they were thinking in terms of 3D cubes but they had not taken into account the fact that cubes would not completely fill in the mug. They actually were having trouble finding a way to make up for the curved walls of the mug. For some reason, thinking in terms of a cylinder did not occur to them. In our dialogue, they did have that 'Ah ha' moment and realized an easier take on the problem. I could tell that they were thinking in terms of volume, and they were keeping in the 3D realm, but breaking a round surface into cubes creates a few problems. The good news was that they recognized something was wrong and were problem solving in how to make up for the missed area, they were just having issues coming up with a clear solution. This kind of outside-the-box thinking makes me happy because even though there was a much easier solution, they didn't give up. They got stuck, but they kept the discussion going and tried new ideas. I believe if I wouldn't have come over, they would've continued working and come up with something that brought them closer to the answer, but not quite a fully correct one.
From this point we moved onto Dan's water tank. After showing Act 1, I actually got the question, "Do we have to watch this thing fill up? Its going to take forever!" I had never done this problem with my students, but thinking back to Dan TED talk, I got the exact reaction he described. Before Act 1 was even over, they were complaining about how long it would take and some actually starting focusing their attention elsewhere. When the video stopped, everyone erupted into, "Hey wait, isn't it going to fill up?" "Uh, Mr. Brandt, the video broke." "I wanted to see how much water their was!" The questions flowed naturally and they got to work. Below is a mashed up, kind of, gross, panoramic-type view of my board when all was over.
I had the students present their solutions because I noticed many different approaches. Throughout their work, I got the question of "How do you calculate the area of an octagon?" numerous times, to which I responded, "I don't know, but I can tell you its a gross formula." Some students wanted to look it up, which I allowed (most were not successful in finding it and those that did couldn't use it correctly). Others broke it into familiar shapes and went from there. Some went for 16 right triangles, some 8 isosceles triangles (both recognized the need for trig without prompting in these processes, which was sweet)., and some broke it into a rectangle and two isosceles trapezoids (which for some reason I never thought of). There was actually only one group that did this last thought process and it came from a group that I did not expect, which added to the awesomeness. Each group explained their thoughts and why they approached it how they did and everyone learned something. I could see on the students' faces surprised looks of enlightenment as other groups were presenting; it was a beautiful thing.
As I explained to my students after this problem, this is what pushed me towards math. I love the fact that every problem can be solved in numerous ways, some more creative than others, but they all take different aspects of math and combine them to form a solution. Its wonderful how all of these numbers and concepts fit together into a nice, complex package that is eerily consistent. This is exactly what I want to push onto my students. I want them to see where it all comes together and to see that any problem can be solved, even if you don't have the knowledge that you think is required. Through these 3 Act problems and allowing the students the run the curriculum through questioning, I believe I'm on my way. After my speech to them, I didn't have any students saying "Yeah, but you're a nerd. You're a math teacher, you have to care about this stuff." Instead, I saw heads nodding in understanding, as if to say, "Yeah, this is kinda cool."
I loved the way this day went. After these two problems, the students saw me project Dan Meyer's google doc list of 3 Acts, and they started requesting them. They were begging me for more problems! As a strict teacher that sticks to the curriculum, I said "Nope." NOT! We did some more! I teach geometry, but we were doing problems that weren't geometry related. Why? Because the desire to learn was present. No one in the room cared that we were moving on to topics outside of our current area of study because they wanted to be perplexed and solve some problems. If this isn't the goal of the teacher, I don't know what is.
All in all, it was a pretty sweet day.