Mr. Steve Leinwand

How many of us teach the way we were taught? How many of us plan lessons relatively quickly because we lecture, or maybe because we teach the same courses year after year and it’s just gotten to be that easy? How many of us observe other teachers for the purpose of collaboration to improve what we do? How many of us believe that if we continue to teach the way we have been, student achievement will go up?

That last question is really what I’m looking at. I know that in my short six years of teaching the same courses I’ve found myself answering positively to the first two questions, but yet negatively to the last two. Unfortunately, I think there are plenty of teachers out there who are not honest with themselves and may believe that what they’re doing is fine and will continue to be satisfactory with the CCSS. This is an issue. I think we can all agree that with the adoption of the Common Core students are going to be expected to do more than they have in the past. Independent thought and critical thinking are going to need to be included in our curricula so they can rise to these challenges. We need to implement strategies and practices into our daily lessons so that we can build up these skills not only in students, but teachers as well. The other day I had the pleasure of listening to Steve Leinwand give a presentation to our IU where he addressed these issues and some actions we can take.

I wish that I could adequately summarize all that he said, but I’m sure I will not do him justice. He started by showing what math used to (and in some cases, still) look like: drill and kill, no context, variables, variables, variables! (He also used this as an opportunity to share his distaste for Algebra 2, but that’s a different discussion) All of this, among other factors, has created little growth, little real-world preparation, and absolutely little preparation for the CCSS math practices. We know this from the math anxiety, illiteracy, poor test scores, tons of remediation, and large amounts of criticism. So… what do we do? The same thing of course (NOPE!). “If we continue to do what we’ve always done, we will continue to get what we’ve always gotten. If, however, what we’ve accepted is no longer acceptable, then we have no choice but to change some of what we do and some of how we do it” (from Steve himself).

He went on and showed all kinds of examples of how we can change, which were all very Dan Meyer-esque: introduce problems with pictures and video, introduce data sets by only giving a few numbers, show pictures, numbers, or representations and ask “What do you see? How do you know? Convince me? Prove it.” All of these tasks followed a similar pattern: show the students just a little bit and let them hypothesize as to what was coming next. The data set he provided was particularly impressive to me because I wish I would’ve thought of it. He showed us a few numbers and had us talk about what patterns we saw, what numbers we thought will fill the rest of the set, and what it represents. Then he showed us a little more and we found a new pattern and took new guesses. Then she showed us a little more and so on and so on. Throughout this process, whenever a new pattern arose, we’d talk about it at length. It wasn’t just, “Nope that’s not right, let’s move on,” it was taking our responses and running with them. It was focusing on the students’ responses, giving them some ownership, and letting them run the class. He did this with every example. He never knew what our responses would be, he didn’t know where we would lead the conversation, but he was always prepared to facilitate a meaningful discussion based on our answers.

As I watched and listened to him, I couldn’t help but think, ‘This is not for everyone.’ I know that I could do that for geometry because I’ve taught it for 6 years, but I probably couldn’t do this for algebra 2 and definitely not for calculus; I’m just not that comfortable with the material. I have a feeling that many teachers would agree with this. So the question is, if the goal is to implement strategies similar to these build the quality of our lessons, how do we build up our teachers so that they can do this? I experienced one option, go to training sessions and presentations like Mr. Leinwand’s. Would another possibility be to allow teachers to teach the same course year after year so that they become comfortable with the material so they can focus more on the teaching strategies and less on the concepts themselves? And obviously, throughout this entire process, there needs to be plenty of follow-up.

That last part is what concerns me the most. Even in my short career I’ve sat through plenty of programs and initiatives in my district that started strong and then fell through within weeks. I know all kinds of strategies that our district bought into, but no one has ever checked to see that I’ve implemented them or that they’ve made a difference among our students. Sure, research shows that certain processes are more effective than others, but if they don’t get implemented what’s the point? This is actually how Mr. Leinwand closed his presentation. He said to the hundreds of teachers that all of this was pointless to 80% of them because they will go back to their classes and continue to do the same thing after being jazzed for a few hours. He called everyone out and no one argued with him because we all knew that he was speaking the truth. We need to be held accountable. The items he discussed would greatly improve the classrooms in my building, in my district, in the state, and in the country. We need to hold ourselves to a higher standard and keep in mind that we need to do what is best for these kids. We need to prove Mr. Leinwand wrong by sharing, supporting, and most of all, taking risks. Even though we’re spread out geographically, with the common core more than ever, we’re all in this together. We’re all teaching the same thing, let’s make sure we’re all teaching it to the same high standard. Let’s collaborate, communicate, and inspire each other to go out on a limb and try something new.

“But… that’s scary. And a lot of work.”

Yes, yes it is. We will need to change, which is never easy. Some of us are stuck in our ways and fear that which is different or refuse to believe any changes will be effective. This also creates a fear of failure – that’s what our colleagues are for. Fear of failure creates lack of confidence. Lack of confidence lends itself to excuses: there’s not enough time, these kids don’t want to learn, they don’t care so why should I, Yeah but… etc. Without proper leadership, there will not be proper accountability or proper support in place. We can overcome these potential setbacks with the proper items in place. We need to envision the possibilities and work towards them rather than work against them. Great things can happen in the right place with the right people.

Mr. Leinwand provided plenty of specific examples of how math instruction can be taken to the next level, and if you’re interested I can try to provide you with some of that information. However, what I summarized above, in my opinion, was the most important part of his presentation. Simply increasing the rigor and relevance of our instruction is easier said than done. In order for it to happen, many other items have to be in place in order to create a supportive network of educators that share a common goal. If anyone believes that they can do it isolated in their own room, they are mistaken. We need to exercise our creativity, take risks, and collaborate for the purpose of increasing the quality of education we provide. Let’s get our students informed, engaged, stimulated, and most of all, challenged. After all, which class would you rather be in?

Special thanks to Steve Leinwand for sharing his insights. 

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?

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. 

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.

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. 

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.

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. 

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)

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...

Sir Ken Robinson

Last night I had the amazing opportunity to watch Sir Ken Robinson speak at Millersville University. I left speechless. If you are unfamiliar with his work, please go here and here and here.
There are so many thoughts running through my head at the moment I'm not exactly sure I can write anything that will do his talk justice. I'm wishing I would've wrote this post last night when I got home instead of sleeping. I do want to highlight just a few points that Sir Ken made that stuck with me and will cause me to rethink my classroom and department.
1. How do you run an organization that is adaptable to change and flexible? One that is creative? One that keeps up with change and stimulates change? Public education needs to be this way. One point that he made, and that I agree with, is that schools are all about conformity. All students take the same classes at the relatively the same time and are expected to get the same grade. Schools need to allow students to explore their interests and creativity so that they can find their element. This will require schools to adapt to the students it educates rather than the students adapting to the schools they attend. There are many steps that need to be taken for this to happen, and its certainly not something that will happen overnight. Can we as individual teachers do anything to support this process, even if the governing bodies do not officially embrace it? Sure.
2. NCLB is actually leaving everyone behind. Standardized testing is causing teachers across the country to mold their students into machines, learning processes but not thinking about what is going on. Usually, this is done in the most boring way possible. When students are looked at as data, they revolt. When they are looked at as individuals, they succeed. A political policy that was supposed to help education and increase our students knowledge is, in reality, taking away from their education because they are being forced to learn about things that they see no value in. They are not able to express their creativity because they are limited to what's on the test. Combine this with the decrease in public education funding and you lose those courses that engage students and stimulate their creativity and you keep the courses that are cut and dry.
3. We are reducing our funding for education increasing our funding for the correctional institutions. 1 in 31 people are in, waiting for sentencing, or being rehabilitated by a correctional facility. Not that those two statistics are directly related, but its interesting to think about.
4. Personalizing education helps people realize their talents. Every attempt to personalize education has failed. Standardized tests de-personalize the educational experience. This was really the subject of his entire talk.
5. By narrowing the curriculum, we are implying that life is linear; that we all will follow the same path. In truth, life is organic; its constantly changing and adapting to surroundings. Let's teach our students how to make these adaptations rather than telling them what to do and where they should be. Let them discover what they are good at and what they are interested in, and let's foster it. If we tell them what to learn and how to learn it and don't move off of the curriculum that is set for 'everyone,' how will they ever learn their place in life? There are plenty of people in the world that are good at what they do, but don't truly enjoy it. Let's have our students graduate ready to pursue a career that they are good at AND love.
6. Myths - 1. Only special people are creative. 2. You are either creative or your not. 3. Special things are required to be creative. 4. You can't teach creativity.
7. As teachers, we are like gardeners and our students are our plants. Gardeners don't grow plants; plants grow themselves. Our job is to provide the optimal conditions for growth. Beautiful analogy.
8. The risk we take in margenalizing our students is greater than the risk of letting them be creative and grow.
9. "I'm not what's happened to me, I'm what I chose to become" - Carl Young

Of course, these are not my original thoughts. They are all from the great Sir Ken Robinson. He said so much more and was extremely informative and insightful, but these are just a few of the points that stuck with me and will guide my classroom from now on.

Mistake?

I was grading some student work today and came upon this mistake.

In the class, we are currently studying problem solving strategies. This particular problem came from the 'Look For A Pattern' lesson. The problem asks the student to find the next four rows in Pascal's Triangle. Now, the majority of my students have not seen or worked with Pascal's Triangle before, so to them this is a seemingly random set of rows of numbers. I did not give them the background knowledge of the Triangle beforehand either because I wanted to see what they came up with as their answer. Usually students get it correct right away. This, however, was new to me. Because I only gave the first five rows, if you look at them as whole numbers rather than individual digits, they are all powers of 11. Pretty neat. This student went with that and continued. Unfortunately, Pascal's Triangle differs from this point on. She did find a pattern, thinking outside the box. Now, I did mention specifically in the problem that it is Pascal's Triangle, but she found a pattern.

Does she have an understanding of finding patterns? I believe so. Do I mark her wrong because her answer is different than what I've got on my answer key? If I'm grading on finding patterns (which I am), then no. If I'm grading on their understanding of Pascal's Triangle (which they may have never seen before), then yes. But then again, why would I grade on something they've never been exposed to?

I thought this solution was interesting; thought I would share.

Lil' Help

I need some guidance, help, advice, etc. The first unit in my geometry class was big on vocab to set the pace for the rest of the course. It focused on all of the different types of angles, polygons, and quadrilaterals. We explored each to their fullest, expanding on and making connections to every characteristic possible. In the past I've given an exam with problems and the students have done fine. Recently I started giving this as a quiz instead...

My students have to find all of the angles that I've labeled (36 of them total) in this regular dodecagon. I give them one right angle, but that's it. They need to find everything else on their own. I like this because they need to use the properties we talked about throughout the unit to arrive at the answers. Its a nice, different type of assessment that gets them to apply their knowledge.

I'm having trouble because I don't know how to grade it. Obviously, if they get one angle wrong that is going to through off other angles as well. I can't take points off for every wrong angle because then I'm potentially subtracting points for the same mistake multiple times, which isn't really fair. I need to find a way that assesses them fairly. I've thought about having them write an explanation of how they arrived at their answers, but that would be a ridiculous amount of writing, even if I made a simpler figure with less angles. I've also though about having them list their answers in the order they calculate them so I can try to follow their thought process, but I'm not sure if that would give me the full picture of their understanding. 

Could I evaluate it in some type of standards-based grading system? Are there any other ways that I could use this to assess my students, or do I just simply not grade it but make it a class activity?

Lil' help?

Note: as I type this I'm listening to the Dark Knight Rises soundtrack and I'm now totally pumped to teach for the day, although my lessons may be slightly darker than usual.

SDC (day 5)

Today a fear of mine happened: I lost track of what I've covered so far in my two geometry classes. I started to plan for the day this morning and couldn't figure it out for the life of me. I prepared some extra material, probably enough for 3 days in reality, but I started both classes with some info we've already discussed. Overall, this wasn't a big deal; a little review never hurts. It kind of messed with the flow we had going in both classes though. Last week the students were directing the curriculum (see previous posts) and then today I found myself moving back to old habits and directing more. I realized it about half-way through my second block but it was tough to get back to the questioning. I found the downfall in this plan that I didn't plan out: when it gets to the end of the unit, there is certain info I need to cover and it gets increasingly difficult to ask questions that will guide the students thought there. And I guess if I'm asking the questions to guide in a specific direction, its not totally student-directed. So, this is going to be tough to organize. I loved the way last week went. The students were into it, I was into it, everything was going awesome. Now, being limited by units, its getting tough as we finish up the first one. Ideally, as I said before, I would like to have the entire course's material planned ahead of time to really let the students run with the material instead of me leading the way. We shall see.
The good news is that I got everything covered in both classes within the same time frame. I had a few students notice that between classes they had differing homework assignments, but it didn't bother anyone.
I did end the day with a lil' bit o' problem solving. Got my students thinking, motivated, and I'm slowly turning them into super geniuses. More on that problem tomorrow...

Student Directed Curriculum (Day 2)

Day 2 of my experimental student curriculum has come and gone, and I must say, it went quite well. Here's how it went down: On the first day I gave them this paper and gave them 5 minutes to calculate as many angle measures as they could as well as find any polygons that jumped out at them.

Obviously, everyone found the right angles, the majority were able to do the vertical angles, and a couple went as far as making the connection to corresponding angles. After the five minutes, I asked for a volunteer to put something they found on the board. Both of my classes started with angles, so I went with that. We started with the angles the students gave me, I asked them how they knew what the measures were, and we connected it to other types of angles and found examples. I was greatly impressed with the discussions we had and reasoning I got from students. This was the first real lesson we've had where I've asked them to explain themselves and they went for it. It was awesome! That was day one.
On day 2, we came back to this same paper, only this time I asked for polygons since we already exhausted all of the angles. This is where it got interesting. I had a student in both classes come up and highlight a few polygons. Both classes pretty much gave me the same list: rectangle, triangle, trapezoid, parallelogram, rhombus, pentagon. I asked where they wanted to start. My second block class voted for the trapezoid; we dove in. I showed them a trapezoid in comparison to an isosceles trapezoid and they commented on what they thought were the differences. We talked about consecutive angles between the bases and related it to corresponding angles. There wasn't a single aspect of trapezoids that we didn't discuss. For the most part, they were able to toss out the ideas and then we explored them; I did very little nudging. We ended the day talking about the reflection symmetry of an isosceles trapezoid, and the relationship to the perpendicular bisector of the bases. I thoroughly enjoyed it.
Then block three came, and oh boy it just got better. Same routine as above, however, they wanted to start with pentagons instead. Game on. The student that highlighted the pentagon was unsure if it really was one because it wasn't regular (only he didn't say regular). Discussion started, we came to a consensus that it doesn't have to be regular to be a pentagon. Great. So I drew a regular pentagon on the board and said, "What's the measure of each of these angles?" *confused/thoughtful looks/blank stares* "Ok, how many degrees are in a pentagon?" 360? 540? 720? Awesome, these students have an idea of where I'm going with this. I drew a triangle. "180 degrees!" someone said. I added a triangle to it and showed them the quadrilateral formed. "360 degrees.?" someone hesitated. So far so good. I add another triangle to form the pentagon. The pattern? "540 degrees!" confidence building. I kept going: another triangle, six sides, 720, another triangle, seven sides, 900, another triangle, eight sides, 1080. What's the pattern? "Oh, I get it!" "How many degrees in a polygon with 100 sides?" "You would take 98 times 180." "Why 98?" "Because the number of triangles is always two less than the number of sides."

"Sah-weet! So back to my original question. What is the measure of each angle in the regular pentagon?" "108. 540 divided by the 5 equal angles." Needless to say, I was excited. Everyone was paying attention. Multiple students were participating (the above conversation included many students). We discussed a few more pentagonal items, and then I guided them toward the symmetrical nature of the polygon. Reflection symmetry was discussed in a similar matter as my previous class, but here I had time to move on to rotational symmetry. No problem. They were good to go even though this was a relatively new concept for them. The only issue I had (or think I'll have) is the phrasing, 'five-fold rotation symmetry.' My students always have trouble with that, knowing what it means, using it correctly, etc. I have yet to find a good way to explain it. Any ideas?
So, yeah, I'm excited about this whole freedom of curriculum concept so far. After my third block class, I realized its going to get tough to keep track of what I taught from day to day. I guess its time I figure out some kind of organization skills past post-it notes.
The students seem to be with me too. Everyone was listening, involved, engaged. Questions were being asked both by me and them, and answers were the same. We're both learning, and I don't think it gets much better than that.
Here's the kicker: I only have this plan 'planned' for these first two units on quadrilaterals and triangles. After that, I'm still trying to figure out how to get it to work. My next units are area/perimeter, surface area/volume, similarity, reflections, proofs, and circles. Theoretically, it would be easy to mix and match units and make all kinds of connections, however, that would almost require me to planned for the ENTIRE SEMESTER since I'll never know what's going to come up in class. Maybe in the spring? Ugh... that'll be tough. We shall see.

Student Directed Curriculum (kind of)

I teach geometry. This is my sixth time teaching it. I think I'm the only person in my department that hasn't changed what they've taught for the past six years. The perks: overall, less planning! The downside: more planning? trying to reinvent the same material so it doesn't seem like the same old thang, while still keeping it effective and relevant for the students.
I always push for questioning in my class. I do stand in front of the class and explain some things, but their questioning fills in some of the holes and also covers the curriculum. In the past, I've followed the set curriculum I helped to write a few years back. Just like any other typical class, I started at unit one and plowed through to unit 13 or 14, following the lessons in the order they appeared. I thought I would try something new this year: let the students decide.
My plan is to start with unit 1, the basics of geometry (terms, notation, polygons) and go from there. I have a paper that I made up with all kinds of lines on it and I provided three or four angle measures. With this, I'll let the students calculate as many angles as they can and find as many polygons as possible in 5 or 10 minutes. From there, they can share what they've discovered and we'll discuss in great detail everything they bring up. For example, if a student give me an angle measure, we'll talk about whether or not its correct and why its calculated that way and how it will lead to other answers. If someone points out a polygon first, say a kite, then we will explore every aspect of kites (angles, segments, symmetry, area, etc.) before we move on. Theoretically, the entire beginning of the course will be based upon this one worksheet. Unfortunately, I don't think I'll be able to give them complete freedom and follow wherever they lead me (not sure its physically possible to be that planned out) (now that I think of it, in order for my students to 'run' the curriculum, you would think it wouldn't require much planning since I'm not doing the work, turns out it might require more, hmmm...).
The downside is that my two geometry classes will potentially be at different points in the curriculum all the time. Its probably going to be tough to keep track of what I taught and what I didn't. The upside is, the students are thinking, their guiding themselves, we're working together, they're engaged. Now, as I type this a question arises, how do I handle assessment? I'll need to provide a certain amount of structure for this to work so it doesn't turn into complete chaos. Do I still keep the unit in the same order so as tests/projects can be used consistently between classes? I realize the exams should not be the motivation for such a decision, but how else would I handle it? If I let the units get criss-crossed and the previous order changes, do I allow it and move to project based assessment instead of exams? Or, do I simply evaluate when enough material is enough and write up new assessments, different for each class?
I probably should've thought this through a little more before the second week of school. Oops.

Nay-Sayin' the Nay-Sayers

This week I've been participating in a 1-2-1 initiative training session in my district. All of our freshmen are getting their own laptops, and its my job incorporate them into my lessons effectively. I'm not a huge fan of technology for technology's sake, but some of the tools we've looked at could add some significance to my lessons. It'll be interesting to play around with it throughout the year. I'm sure I'll post an update in the future.
Anyway, this morning we got into a discussion on project-based learning versus the 'old style' of education. The obvious question of 'Which is better?' was posed, and everyone agreed that the PBL style would be much more effective. When our presenter, Tom Gaffey from Philadelphia School of the Future, asked how many of us have tried it, less people raised their hands. When he asked why we haven't, what our roadblocks were, this is where the discussion got started. Many of us began talking about pros and cons to planning and executing these lessons, and the end product was very positive.

Now, I'm in a unique situation in that I've taught the exact same courses for the past six years. I've tried to use this to my advantage. Rather than teaching the same thing every year, I've tried to add or modify a PBL unit to the course every time I go through it. 

Whenever I take part in these discussions, I often get the feeling that teachers really do want what's best for their students, and these presentations motivate them to put the proper pieces in place. But once the show is over and the presenter is gone, temptation to resume the status quo emerges. It took me many years to overcome this feeling, and it still lurks at time. I understand that the time factor in planning is tough, but that is why you start small. Lack of motivation among students is present in every math class (it's what we do!), but PBL lessons have a 74% chance of generating more interest among students (I made that up, but they will help!). State exams and standards aren't going anywhere, if anything their gaining momentum, but we need to work with them not for them. This is the excuse that bothers me the most, and I know there are people that will disagree here.

The example we looked at today was teaching slope through building a set of stairs. Mr. Gaffey explained how he had his students find someone in the community that needed a set of stairs built and his students did the work for them. He taught them everything they needed to know to construct them correctly and tied the notion of slope into his explanation (I immediately thought of Dan Meyer's competition idea). To me, this is an awesome idea! He even admitted that his students did no better on the state tests than others. At this point, the conversation got interesting. I could tell some in the room began questioning the significance of the lesson. After all, if it doesn't raise scores, but takes more time, what's the point? I've got stuff to cover, we've got to push through the content. On the contrary, Mr. Gaffey made the point that his students gained a valuable learning experience in problem solving and construction among other things. He took it from concrete to abstract and had the students attention the entire time. So, yeah, it took longer, but the benefits far outweigh that. I would love to have an entire course (or all courses) structured in this way. If I get through the entire curriculum, GREAT! If I don't, who cares? If my students learn the majority of a curriculum really well along with other topics, then whatever I miss can be made up later. If my students have the opportunity to get a ridiculously awesome learning experience, I'm going to do anything I can to make that happen. It's not easy, but then again, if it was everyone would be a teacher, right?

I wish I could find more educators that shared this mindset and had a better opportunity to collaborate for this type of planning. I believe great things can happen if education is done right. There are many things that work for our students, but we need to ask ourselves if what we are doing is whats best for them.

With A 'Lil Bit O' Algebra

So, I've recently been asked to rewrite our districts Algebra 1 curriculum (and eventually Algebra 2 I believe) to fit with the Common Core State Standards, again (woot!). Teaching something because the standards tell me to bothers me as opposed to teaching something because its relevant, but that's a rant for another time.
A colleague of mine and I were looking over the CCSS for alg. 1 and realized its kind of a hodge-podge of topics thrown together. I mean, all of the linear equation/function stuff works together very well, but then there is some beginning stats/probability and also rational expressions, polynomials, exponents, etc. thrown in as well. We were trying to find a way to organize this course so that it makes sense and there are logical transitions. As of now, the topics are taught in the order in which they appear in our textbook (UCSMP), which is no good (both the organization and the book). We stared for a while and threw out some ideas, and then I realized something. Every alg 1 curriculum that I've ever seen has always ended with stats topics, and they are part of the 'if there is time' category. I wanted to give this course some flow and a context, so I thought 'Why not teach it from a statistics perspective?" After looking over the standards again, we figured out that this just might work. Here's the tentative plan:
      We'll start off with calculating different types of probability. This covers the different types of numbers, how to order them, represent them, compare them, etc. We will then move on to different ways to represent data, bar graphs, pie charts, stem and leaf, box and whisker, etc., further emphasizing the importance of number sense. This leads nicely into scatterplots and line of best fit, which opens a door to teach all of the linear equation/function topics that are essential to an algebra 1 course. This is obviously a very loose description since I don't have all the info in front me, but I think it will work. Every topic will have a context and we'll be able to teach everything in a real setting. My hope is that this allows students to see how these can be used and provide an easy method to be taught. I'm very excited for this to happen and can't wait for the results. It makes me wonder why I've never heard of anything like this before.
      The only hicup - the rational expressions, polynomials, GCF, LCM, exponents topics that are to be included. How do we incorporate them into a stats context that flows well with everything else in the course? As we talked, the best we could come up with is 'throw them in at the end.' No context, no transition, no meaning in regards to the rest of the course. This upsets me, but, I've got nothin'. Any ideas?
I feel that this is a new and exciting way to teach algebra 1 that could produce some amazing results. I'm a little nervous about showing this to those who are teaching alg. 1 next year because its so different than the way it used to be, and also because I'm not teaching it (so that will produce some interesting discussions as well).

Snow, man.

Checked out my 101qs submissions today to see the low perplexity scores. After checking up on Dan Meyer's blog I realized that my 'Snow, Man' pic had an honorable mention in his Top 5 for this week. I won't lie, I got excited. However, after reading his response to it, I realized that I had missed the mark slightly. My picture gives way to some basic questions, but could be better if planned out.

I would imagine if I showed this picture to my students I'd get questions like, "How much snow is that? What is the ratio between the three balls? How long until it melts?" and so on. These aren't bad, but are they really any better than a textbook problem? After all, the whole point of Act 1 is to get the students engaged and really interacting with the math behind the situation. I need something that will stimulate independent thought and make the students care; something that will allow them to envision themselves in it. Everyone builds snowmen, but who really cares about how much snow they've used to build it?

After reading the comments about how to redesign the problem, I've got some ideas to reproduce this in a few months. I liked the idea of comparing it to another snowman in a neighboring yard - introduce the competition element (after all, everything's better when there is a winner and loser). I could start by showing an aerial view of my yard along with a time-lapse video of me building the snowman in a methodical way, clearly using a certain amount of my yard. I could then pan out showing my yard again with the amount of snow that's been used already and the amount left. 

This could still produce the same questions as before, but now it could be expanded to "Who's snowman is bigger? What is the biggest possible snowman that could be built? How long would it take to build such a snowman? Could you combine the two to make a super snowman?' While these questions may not seen anymore advanced or intense, the redesign could get my students involved more. It wouldn't have as much of a 'ugghh, not this again' context, but would be more inviting. It would be more practical, more relevant, more realistic to their lives. Getting them engaged and thinking is the key, and this might just do it. I'll let you know in 9 months.

This has made me realize that I still have some work to do in creating these First Acts. While it can be done with almost any picture or video that inspires thought, they key is using the best possible option and format. Taking a basic picture like that above works, but does it work as well as I think? I need to think about these from a students' perspective and a teacher's perspective to get the best possible Act. Practice, practice, practice. Thank you 101qs and Mr. Meyer!

Order Matters at 5-Below

I was walking through 5-Below this afternoon looking for some fun stuff for my daughter and saw this hanging on the wall: a customizable iPod case with 'over 17,000 possible combinations!' Whenever I see advertisements like this I always wonder if they have a math guy that figured that out for them or if they just estimate it. I also wonder how many people will change the way their iPod looks 17,000 times.
If I showed this to my students while teaching permutations and combinations, which would they argue that it is? Is this an engaging enough picture to get them to think about it, or would they quickly move on?

Oh, the possibilities...

Any (mathematical) Questions?

Piece of Cake Upside Down

I recently re-watched one of the few good math movies made - 'Stand and Deliver.' I showed it to my students and they were (surprisingly) interested in it and loved it. Some of them learned shortcuts to their 9 times tables and a new way to think about positive and negative numbers. I, on the other hand, watched it from a new point of view.

I haven't watched this movie since I've become a teacher. Previously, I watched it from a student's perspective and purely for entertainment value. Now, as a teacher, I realize there is some deeper content here. There are some things wrong with the clip above. The claim that their students cannot learn because of where they live or because of their status is bogus. Mr. Escalante gets it right when he says 'students will rise to meet your expectations.' I think there are a good amount of teachers out there that don't fully believe this; they think kids will be kids and there's nothing that can be done to help them. They are who they are and there are all kinds of excuses for the teachers not to teach them. Escalante owns up to his responsibility, says he could do more, and he follows through with it. After watching this again, I began thinking about my classroom. Do I set a high level of expectation for my students, and continue that expectation throughout the semester, or do I eventually cater to their level? Can I get them to do more for me and for themselves in order to help them realize their true potential? Is there a way for me to give them the ganas they need to be successful not only in my class, but in their lives? If I answer in the negative to any of these questions, what can I do to change? I plan on seriously reconsidering my first week of class and my management techniques to empower my students with ganas. Yes, I teach math. Yes, its an uphill battle before they even walk in the door on the first day because of that. Yes, stereotypes say that I am a boring nerd. I don't think that I can settle for any of that and I don't want my students to either. I need to find a way to change their opinions, change their mindset, and change their opinion of their ability level. I need to gain their trust from day one and show them that they are capable of whatever they put their mind to. I feel I do this to a point with some students, but not as many as I'd like. I also realize that this is a Hollywood interpretation to a true story, but there is a lot to be said for it. I could go on and on...