## Tuesday, November 26, 2013

### #SoPowerful

Last year I posted about this diagram (http://pabrandt06.blogspot.com/2013/05/why-i-love-trig.html), but at the time I was not [technically supposed to be] teaching trig of any sort. The only material that was in our curriculum was right triangle stuff in geometry. I through some of the more in-depth trig in at the end of the semester because I felt it was incredible, but I had to neglect some of the topics in order to do so. I also did not spend enough time on it for the students to fully grasp the awesomeness that was happening.
This year, I'm teaching pre-calculus for the first time, and I'm taking full advantage of being able to teach trigonometry as deep as possible. We're spending the entire second half of the course on it, and so far after about 4 weeks, my students have still not picked up a calculator, and they know an tremendous amount of material. They are getting a great depth of knowledge on this topic, and they are making all of the connections without any kind of calculating device whatsoever. To me, that is awesome. Not only do they know the unit circle inside and out, but they are also beginning to estimate values of the trig functions as well.
Our ultimate goal was to learn the diagram above, which has been achieved. This diagram gives a beautiful illustration of where the names 'sine,' 'cosine,' and 'tangent' come from, a great visualization of where the trig functions fall within the unit circle, and a natural derivation of the Pythagorean Identities. If students can learn this diagram and understand what it means, they have gained a ton of knowledge that I'm willing to bet is not necessarily taught in the average trig class. I have not seen this in any text book (unfortunately, most text books don't seem to focus on the unit circle, which I don't understand), maybe this is above high school level, but I do believe it is ridiculously valuable.
Our next step is to use it to graph each of the trig functions. A colleague of mine has created a Sketchpad file where the coordinate rotates around the unit circle and all six graphs are created simultaneously. It's a little overwhelming, but captivating and artistic at the same time. When he showed it to his class, they were able to match each function with the appropriate graph, and they now understand why the graphs look the way they do. They were also able to explore the relationships between each of the graphs, why the asymptotes exist, and so much more.
I'm not sure if the half-angle and double-angle properties are in there (I haven't investigated that quite yet), but I'm hoping I can come up with something.
All in all, this diagram excites me. It makes sense, and it allows students to have fun with numbers and they don't question it. I have not had a single student refuse to do any of this work. They see the connections, they appreciate the math that we are doing, and they understand how much smarter they are now.

## Monday, July 15, 2013

### Seeking Feedback on a New Idea

This year I'm going to get to go through a trial run of something new in our high school. As far as I know, it hasn't been done before in any school around here; it actually is something you'd see more in a college setting. I will be team teaching algebra 2 and pre-calc with another teacher. This isn't co-teaching in a sense that most teachers think about - I will be teaching these classes with another math teacher. Essentially what will happen is we'll combine our two classes (50-60 students in one room with two math certified teachers). My colleague and I are excited about this opportunity and knowing that it's never been done before, we are expecting both successes and failures as we go through this process.
We started teaching across the hall from each other this year, and because of this, started sharing what we were doing in our classes. If I thought something went great, I'd share my excitement. If something went poorly, I'd ask his advice. He did the same. In this collaboration we realized that we had very similar teaching styles. We both have a desire to push our students to their highest level through problem solving. We also want our students to understand why all of this math works. The 'why' is the biggest part; we stress that more than anything. All of this produced the idea of teaching together. We were curious to see what we can do together in the same room bouncing ideas off of each other as the class progresses.
We tried this is short bursts last year. We would poke our heads into each others' rooms during our planning and just jump in the lesson, adding our two cents when appropriate. It worked great because we were able to think of things that the other didn't. The students never knew who was going to speak (we didn't either) and because it was new to them, they were hooked. After doing it randomly without planning at all throughout the year, we asked the students which they liked better - one or two teachers. Hands down they all preferred the two teacher model. They liked the "structure" and they liked getting ideas and info from different perspectives. They also liked being able to get individual help. After these few lessons were assessed, their achievement was up (not that there was a cause and effect relationship).
Together we loved teaching this way. I think we were able to learn from each other as much as the students were learning from us. Obviously we saw great potential in this style which is why we are pursuing it on a grander scale.
However, I write this not to share my experience so much, but to ask a favor. We want to make this as beneficial for our students as possible. So I ask you: If you had the opportunity to teach with someone that you really worked well with (whether its in your building, or a fellow tweep, some make believe person you have yet to find, or a body double of yourself... or Dan Meyer) everyday, what are some things you would do in the classroom? How would you organize it? What could you do with two math teachers in 90 minutes for 50 students? We will be teaching algebra 2 and pre-calc, are there any specific topics that you could do some really beneficial stuff with? What are some challenges you foresee? How should assessment be handled? Literally, what are any thoughts you have with this? Do you think it will be a disaster, or do we have a chance?

## Friday, May 31, 2013

### Why I Love Trig

I've always loved trigonometry; it just makes sense to me. In high school I remember it being one of the first times where the math made perfect sense. There are so many connections that can be made between topics, and everything fits together so beautifully. It really is one area of math where you can do some pretty cool things rather easily. I find that even if I don't make 'real world connections,' students are still interested in learning it. Because the connections are so easy to make, and some of those awesome things are easy to derive, students are willing to jump on board and have their mind blown.
Now, with all of that being said, I've never actually taught a pre-calculus course where the majority of trig. is taught. After I graduated I had a long-term sub position where I taught an advanced pre-calc, but as I look back I can't really consider what I did teaching. I hereby issue a formal apology to the students I had back then: "I apologize for not showing all of the awesome things you can do and explore with trig. I failed you as a teacher and I hope that you had someone else in your future that was able to clean up the mess that provided you."
Next year I have the pleasure of teaching pre-calc (I'll actually be team teaching it, which is even better) and I am extremely excited about it. I've been talking to my colleague that I will be teaching alongside about the curriculum and we've developed some pretty exciting stuff. Our goal is to get our academic students to demonstrate some high quality thinking and problem solving. Everything we're going to do next year with trig. will be based off of the unit circle. The plan is to spend enough time developing the unit circle and then use it to create the graphs of the trig functions and go from there (obviously this is a real brief description of what we'll be doing throughout the course). The diagram below is a goal of ours. If we can get our students to develop this diagram on their own and fully understand what it represents, we're golden.
This diagram pulls everything together in mind. It shows the definitions of sine and cosine, it illustrates the meaning of sine ('half chord') and cosine ('complement of sine'), it relates the tangent function to a tangent line (which students often think are two different things with the same name), it gives a visual representation of secant, cosecant, and cotangent, and each triangle gives the three Pythagorean Identities. Wow! And (hopefully) if we can set the correct groundwork with our students we'll be able to get them to generate this and know exactly what's happening. There will be no memorizing formulas or values, just pure understanding.
I could go on and on, but I will save that for when I'm done actually teaching lessons.
Wish us luck!

## Tuesday, April 16, 2013

### For Real, Though. What's The Deal?

I realize that this post is a little late, and Mrs. Fawn Nguyen has already beaten me to the punch, but I'll go for it anyway.
Recently, Mr. Dan Meyer posted about an interview with Sal Khan in which this question was posted: 'What makes sports practice satisfying and how is sports practice different from math practice?' When questions like this are posed, my first instinct is to go to the source - the students. Since they are the ones making the decision, it would make sense that they could give us a straight answer right? I decided to put this to the test and ask some of my students. Below are some of the responses.

"I think kids are more up for working on sports rather than school because kids can actually see a physical change in their lives. more than just seeing a math problem and it being easier. plus you sometimes control how much more change you can see when its in sports."

"Alright, so this entire block I have been processing the question imposed with some deep thought. Yes, I do indeed have a brain. Shocker, right?! I think the answer that most teens would say is that you do what you are interested in. My personality is a good example! I don't enjoy participating in an activity or putting a great amount of effort into something that doesn't interest me. (It doesn't have to benefit myself for me to do it though. I ain't about that selfish life ;p) SO! Bringing this back to the question, sports practice is satisfying due to the fact that the student selected to do it, and it is normally a passion of the kid."

"Its definately an interesting article. The content is all correct unfortunately and I honestly don't know why I see it like that myself. My only theory on the issue is that it's easier to physically exert yourself than mentally exert yourself. Or in other words you would rather run 3 miles than solve a basic algebraic equation. There are 2 kinds of people, the ones who are scholars and the ones who are athletes, and sometimes there is a good mix. But, do you have a passion for Math or do you have a passion for Baseball? Would you rather get your Ph.d in formulas and theorems or would you rather get drafted into the Major Leagues? It comes down to passion, does that make any sense? It's either your passion or your just a whiner. XD Really interesting Mr. Brandt thanks for the read!"

And one more...

"In regards to your inquiry, I am taking my quite valuable time to respond. In all due respect, my first thought was that most math taught is not applicable in the everyday life and thus a waste of time. However, upon pondering your attached article, I realized that my previous cognition was not valid. Math is an essential quality of our lives. We use it with out even giving it a second speculation. Though I have stated one of my opinions, I, in turn, have not addressed your main question.
Sports practice and math class differ in that one is an extracurricular activity while the other is an intellectually involved course. The descriptions above only graze the surface of the in depth reasoning to sports practice being more enjoyable than a math class. Sports, are a freedom that you choose to participate in that differs from math which you are required to partake in. With that statement, it is my belief that many students as well as me, feel that because math is mandatory we are derogatory towards it. We didn't sign up for it so why would we put forth an effort.
In another context, to me sports practices are more satisfying due to the belief that participants get more out of it. On the other hand, their is no way to measure these two oppositions to accurately compare them thus rendering my above statement vulnerable to critique. Sports are a great activity in which to get in shape, become a team worker, and learn beneficial life lessons. However, math provides us with knowledge and the ability to use that intelligence to succeed in the real world. Whom is to say that one is better than the other? While students tend to lean towards sports they must also realize the importance of mathematics.
So, to wrap up, students would rather participate in sports because they think they get more out of it and it is "fun". Were as math is a pointless burden to them"

So, ideas - instant reward, interest, passion. Now I did not send this to all of my students, so this isn't exactly a good sample. I chose students that I have this year that are dedicated athletes and in good academic standing and that I thought would take this seriously. I sent it out to about fifteen different kids, but only received back four responses, so there is that too.
To be honest, this is about what I expected to hear back. Everyone gravitates towards their passion and interests. Even if they don't have that part of their life figured out, they will figure it out by moving towards what they find fascinating. Also, as we know, students are all about their immediate satisfaction. Heck, I'm the same way. I've gotten better with age, but it is still difficult for me to stay motivated on something when the only goal is long-term; even in those cases I try to get it done quickly so I can see the end result.
I was surprised, however, to see the slight confusion in their responses. Even the students themselves are not 100% sure as to why they gravitate to sports instead of school. Perhaps it is something that they haven't really thought about themselves but it just kind of happens. I'm sure if I got all responses back there would be more reasons based on the students' personal connections as well as some reinforcement of the above.
In the last response, I found the phrase "...because math is mandatory we are derogatory towards it" interesting. Is this a reflection on how education is organized? Rather than educators seeking students' interests and teaching the content through those avenues, we are teaching content because of state tests. We are assigning grades based on 'student performance' instead of assessing them based on true mastery, which causes them to care solely about the grade and not about the content. If students had the option of taking various math courses based on where they saw themselves in the future, would the enjoyment factor go up? If art students took geometry because it's applicable but not algebra 2, would we see an increase in interest in math? Or is there still an overarching feeling that math is lame, difficult, and useless that will never go away?
I could continue to analyze these responses, but I am curious what everyone else thinks. Leave some comments, provide some insight. If I get any more responses, I will post them.

## Thursday, April 4, 2013

### Seeing Things Differently

I hang out with English teachers too much; I find myself searching for meaning in everyday events when before I just took them at face value.

Yesterday I was speaking to a colleague about how teaching is changing, not with technology, but with more exploring and student directed discussions. When I read about successful lessons/teachers, they are always ones that give students freedom and ownership, very student-centered. These lessons show how the math works, where it comes from, how it's connected to other areas, and what cool things can be done with it.

I went to school where the model was teach, examples, practice. Because of this, I grew up being a terrible independent thinker and problem solver; I had to teach myself these qualities in college. I graduated high school being able to only complete problems that I've seen carbon copy examples of. I do not wish this experience for any of my students, and I believe many good teachers agree.

I had to learn (and still am, everyday) how to organize lessons focused on my students on my own. My undergrad work too was more teacher-centered than I would've liked, as was my student teaching, and there was never any criticism for it. It makes me wonder how colleges are training future teachers. How they changed with current, effective practices or are they still training in the same old fashioned way?

Our conversation continued with this and we ended it being positive about the future of teaching practices and slightly negative about some current practices. I left to go make some copies and in the machine I found a copy of the poem 'When I Heard The Learn'd Astronomer.' I've heard this before but never thought about it in depth. However, after the conversation I just participated in, I stopped and reflected. This poem describes what's currently going on in math education. We've got some really knowledgeable teachers that are showing facts and step-by-step processes, but they are not effective because they are not engaging their students. There are no connections, nothing interesting for non-mathematicians to grab on to, just straight facts. Math has a stereotype of being challenging, boring, and just for nerds, and that's really not true at all. Just like any subject it can be accessible to all students as long as it's presented in the right way.

I often read through my twitter feed and various blogs and wonder what it would be like to teach in a school with everyone I follow. I'm constantly reading so many great ideas from teachers that get it and are constantly striving for perfection and challenging themselves. I wonder how successful a math department of this caliber could be. I would love to participate in professional development opportunities with them to engage in conversations that are longer than 140 characters. To work with teachers that share this common goal, that is the dream. When everyone's heart is truly at the right place and they are doing what is best for their students, mastery happens, and with that independent interest grows. And that is where the learning occurs.

When I heard the learn'd astronomer;
When the proofs, the figures, were ranged in columns before me;
When I was shown the charts and the diagrams, to add, divide, and measure them;
When I, sitting, heard the astronomer, where he lectured with much applause in the lecture-room,
How soon, unaccountable, I became tired and sick;
Till rising and gliding out, I wander'd off by myself,
In the mystical moist night-air, and from time to time,
Look'd up in perfect silence at the stars.

- Walt Whitman
Leaves of Grass, 1900
emphasis mine

## Tuesday, March 5, 2013

### Quadrillions of Pennies?

The other day, a colleague of mine and myself were discussing exponential growth and how to open this to his students. I've never taught it before, so I felt that I wasn't the best person to ask but thought I could learn something as well. He suggested the classic problem of 'would you rather have \$10,000 now or one penny today, and twice as many pennies each day for a month.' I thought this would be great, and I mentioned the similar problem of if you have a bean on the first square of a checkerboard and you double the number of beans for each square, how many will you have on the sixty fourth square. He decided to go with that instead, but use pennies instead of beans. Long story short, his students were amazed at the fact that there would be 92,233,720,368,547,800 pennies on the last square, not to mention the total amount of 184,467,440,737,095,000 pennies on the board.
Needless to say, both of us were also shocked. In my geometry classes, I've been trying to take our calculated values and putting them into a context that students can relate to. Saying that a box has a volume of 150 cubic feet means nothing to a student until they can see what one cubic foot looks like. So having a value so large was just that to his students: a large, inconceivable number. We tried to put it into a context that was relatable but we kept coming up short. Nothing that we did, relating it to the dollar amount, distance, etc. made sense to us or our students. We came up with one good comparison, but it still wasn't the best.
I decided to use this as an opportunity to experiment. As we were discussing the number I had students in my room taking a test. We were speaking just about the numbers but did not mention the context. One student looked at me afterward and laughed saying, "You guys are such nerds, talking about big numbers." I went on and on about how cool it was for the answer to be that big, and then I realized he had no idea what I was talking about. I described the original problem to him, and he became interested. I then put it into context, saying that's like if you stacked pennies on top of each other, that stack would reach Pluto from Earth 1,911 times. His response: "Wow," and then he pondered. I could see the look on his face, focused on what I just described to him.
Subject #2 conversation went like this - Student: "Mr. B what's this large number on your board?" Me: "Go ask Mr. Miller, he'll tell you." Student comes back: "Yeah, he said something about pennies and checkerboard. I wasn't really paying attention." Me: *describes problem* "That's how many pennies would be on the last square." Student: "Ok." Me: "That's like if you stacked all of the pennies on top of each other, that stack would reach Pluto." Student: "Woah. That's awesome." Me: "955 times!!" Student: *mind blown*
I type this because context matters. I've tried this discussion with many other students and the response was always the same: they start out not caring but then are super interested when they are able to relate to what is going on. Now, I understand that there are some math problems and topics that can be engaging in the pure form that they are. I'm not saying that everything has to be taught in a real-world context. However, students need to be engaged and they need to have a reason to be engaged. I do not like when entire courses are taught without any context whatsoever. How does this help students? In my opinion, this helps to create students' hatred of math. They don't have a reason to care about it so they don't try, which results in them not succeeding, which results in anxiety, and the cycle continues (well, perhaps that's an exaggeration, but it certainly does not help). I want everything I teach to be engaging and relatable, whether its connected to the real-world or previous topics we've discussed. Nothing should be so abstract that they can't comprehend what's really happening. I believe math can be interesting to all students, we just need to figure out how to get them hooked from the beginning and everything we can to keep them there.

## Monday, March 4, 2013

### Up The Banana Tree...

As I've discussed previously, I've been trying to incorporate much more higher level thinking and processing into my lessons this year (as I think most teachers have been). My style has become one where I present a question or scenario to the class and together we build the necessary concepts from it. Five years ago, this style would've scared me to death; I would not have been able to give my students that much freedom. I was not comfortable enough with the material nor was I confident enough in my teaching abilities. However, now that I've taught geometry multiple times per year for six years, I'm pretty much at the place where I could walk in every morning and just go without missing a beat. As a student teacher I remember of dreaming about getting to this point in my career so I wouldn't have to plan much, but now I'm realizing this is not a place in my career that I want to be in. I'm getting tired of teaching the same topics over and over again. I wouldn't say that I'm getting burnt out, but its getting more and more difficult to come up with some new ways of teaching these same topics.
I've been reading some recent blog posts lately about challenging myself as a teacher, and while I don't think I'm at the peak of my career or abilities by any means (there's a ton that I could improve upon), I was able to relate to these posts in terms of my scenario. Teaching these same topics is challenging to me because I've taught them all in multiple ways and I can't think of anything new. I'm starting to get bored but I do not want to be. This new style has forced me to think outside the box and caused me to come up with some really neat things for my students.
Every year when I get to the area/perimeter topics, I wonder if it is really worth exploring. I mean, really? Teaching honors high school students area/perimeter seems unnecessary. For this year, I started with Pizza Doubler and just let the students discuss. It's an easy problem to get into, but one that gets real in-depth real quick. Without any prompting, just with me projecting the picture, my students immediately started talking to each other about it. Listening to their conversations was awesome; so many great, in-depth thoughts and arguments. From their talks, I asked for their estimations/reasoning and we went right into the calculations. They gave me different ideas as to how to calculate the area of the sector and arc length and we tested each one. Their explanations allowed the rest of the class to see their thought processes and we figured out which way seemed to make the most sense. Both of my classes came up with four or five different ways of making the calculations, and still a week later, I have different students doing different things (which is awesome!). They are all doing whatever makes sense to them. They are not begging me for formulas or getting mixed up with the algebra - they fully understand what's going and they have internalized it well. Seeing this makes me excited on so many levels. To be honest, just to see what would happen, after all of this occurred I gave them the formulas that were in the book. I wanted to know if my students would go back to their old ways of regurgitating with formulas or if they'd work with what they'd developed. They did not fall into the trap. Actually, some of them told me that the formula didn't make a lot of sense to them and the way they thought of it worked better. Obviously I didn't fight them on this and had a little victory party after class was over.
Anyway, as we went through the calculations for area of a sector and then for arc length, the students were instantly intrigued as to why the area multiplied by four and the arc length only doubled. They were shocked and perplexed and instantly began asking questions and hypothesizing. By answering their questions, we turned a discussion on area/perimeter into a discussion on similarity. They were listening out of pure curiosity, not because it was going to be on the test. I had them hooked and they wanted more. When I finished my explanation, they sat quiet for a few seconds just taking in everything that they had seen. Breaking the silence was a student with the quote of the year: "We went up the banana tree and found oranges." Absolutely, yes we did! I don't think students are seeing these connections being made within their other math classes, and that bothers me. Looking at their faces throughout this lesson, I could tell that they were captivated with how all of these things tied together, and how predictions could be made. And all of this came from just a (seemingly) simple pizza problem!
I've seen and worked with teachers that are not able to give up that much freedom in their lessons. Had students asked them about similarity during area, they would've said, "We'll come back to that in a few weeks, don't forget your question." Or, I've been with teachers that start with vocabulary, then example one, example two, example three, then their students do twenty identical problems that are carefully made to look just like what they did in class and nothing more. I need my students to have ownership. I need my students to be constantly thinking about what they can do with the math in front of them. I need my students to take the basics to the next level. I need them to understand that all of this 'math stuff' is interconnected in a beautiful way. This lesson achieved that, and I desperately want to develop more lessons that do it too.
I do not know what courses I'll be teaching next year, and even though I've been able to find a new avenue to explore geometry, I would like to try my hand at something new. Regardless of whatever I teach, I'm going to approach it in this style from day one. I want my entire department to approach their lessons in this style so the students in our school develop mathematical thinking abilities and critical problem solving skills. I want to hear meaningful discussions among our students as I walk down the math wing and I want them to get excited about being successful because they've done something important to get there. I don't think that this is too much to ask. As I've stated before, it might be an adjustment for some teachers; it might be way out of their comfort zone. But, if you're not doing something that challenges you, are you really doing everything that you could be?

## Thursday, February 21, 2013

### Ugh, What Now?

I presented this problem to my 9th grade honors geometry students, and we got stuck. While we were working through it there was a retired math teacher in the room as well who was also stumped. After class, I showed it to a few of my colleagues; also stumped. We figured out a few possible paths that might help, but we are trying to solve it strictly within the constraints I gave my students, using knowledge that they would be able to grasp.
We've been working with all of the different types of angles, quadrilaterals, triangles, and all of their properties recently. We've stumbled upon regular figures and what we can do with them. The end goal is to be able to do this, but we've been going through similar problems as a class to lead up to that point.
Here's what I gave them today (please ignore my phone shadow):
I gave my students a regular hexagon, drew in some lines and labeled some angles to be calculated. I love these problems because it takes all of the knowledge that they've gained over the past few weeks and puts it into one beautiful package. They have to know the types of angles, symmetry, properties of various figures, etc. to solve for all of these angles. It's awesome to watch their brains working and the excitement on their faces when they figure these out. Even students who ask, "Why do I need to know how to do this?" are still engaged and have a desire to figure it all out. But anyway...
In this problem, the black numbers are the names of the angles and the blue numbers are their measures. If you look carefully, we've calculated all of the angles that I drew in, whether or not they're labeled. Except for angles 14, 15, 24, and 25 at the top.That one line that passes by all of them (connecting the upper-left vertex to the midpoint of the other side on the right) is really messing things up.
To start the problem I gave the students no angle measures; they had to calculate all this by themselves by figuring out a possible starting point. It is a regular hexagon, and it is drawn to scale, but that's all the info they had. I did not give them any side lengths either since we were focused on just finding the angles.
Angles 14, 15, 24, and 25 really threw us off. We've tried extending some lines outside the hexagon, we've tried drawing extra lines in, and I even had a student line up a congruent hexagon along the side formed by angles 6 and 16 to see if that would help her. My next step is to give the sides a length and see if trig. will lead me to freedom, but I was hoping to get the answer using only angle measures.
Can you figure out the measures of angles 14, 15, 24, and 25 using only knowledge of angles, polygons, and symmetry? Or is more information required? If so, what else do you need? Any guidance would be greatly appreciated.
----------------------------------------------------------------------
**Note: I did figure this out, but it required me to work with the side lengths, using law of sines and cosines. Theoretically, my students could do that if they recognize the fact that the length of the sides doesn't matter. I've taught them law of sines/cosines, and I have some really bright students that could possibly make that connection. However, I am still curious to see if there is a way to do it without referencing the sides and using only angle measures. I'm leaning towards 'no, it can't be done' but still not fully convinced.

## Wednesday, February 13, 2013

### Am I Really Making A Difference?

The other day I was talking to a colleague of mine and the following question came up: 'Does what we do really make a difference within our students?' This question, while it was supposed to be part of a short conversation, quickly turned into a meaningful discussion that has got me thinking about what I do in the classroom and the true, long lasting effect I have on my students (if any).

The department member that I was engaged in a conversation with has a very similar teaching style to my own, one of which I believe is gaining popularity in the math education world - a style of inquiry-based, student-centered education. We both pose challenging problems to our students and use them to investigate new topics; keeping the students engaged with the material by giving them tasks that are just out of their reach, keeping them thirsty, and illustrating the connections that exist within mathematics. We've both had great success with this style, and its apparent by the students' comments and interest that traditionally has not been seen in past math classrooms. Throughout my department, I am trying to push this style and encourage my fellow colleagues to step out of their comfort zone and give their students some freedom and control of the classroom. The times that I have heard of them trying this, they have reported success, but I'll be honest, as the department facilitator I'm not entirely sure as to how much this is happening in our classes. Is it occurring on a regular basis? Are the majority of my department members doing this? Do most students see standard, old fashioned lecture-style lessons straight from a textbook throughout the majority of the high school careers? I have my thoughts, but nothing based on fact. This 'unknowingness' tells me that I need to get into other classrooms more. I need to observe what's happening in my department so that if our students aren't achieving what they should, I can locate any possible issues. With these thoughts running through my head, and with the conversation I had yesterday, I've been wondering if my students really are different at the end of a semester with me, and if they are, does that change last or get reset?
In talking with my students over the years, they have mentioned that they enjoy my class and my teaching style because they see the connections and are forced to work to their potential; they learn to problem solve instead of regurgitate and they become critical thinkers instead of machines (well, most of them). I can see this develop in them throughout the semester. As far as concrete, data driven evidence, I'm not sure I have any, but I can witness the growth. My fear is that when they move from my geometry class to algebra 2, do they become the student they used to be, or do they continue to approach math in a new way? If their new teacher does not challenge them, do they lose that ability to think for themselves? And if so, can they bounce back if they get a teacher that can push them, or do they need to be 'retrained' (I don't really like that word for this context, but can't think of anything better)?
Ultimately this conversation led to us being fearful that our students go back to their old habits and what we've done with them is almost a waste. I hate to sound so negative, because I certainly do not think that what I do with my students is a waste, but if there are no long term benefits then I don't know how else to think about it. Throughout the semester I try to build my students up to a point where they are not afraid to attempt any problem and they can be proud of the work they complete, even if its wrong. If they move on to a class where all of the information is given to them directly and all they need to do to be successful is follow an example in a textbook, then they are not being challenged and they realize that their work is not valued. They may not have to try really hard throughout the entire semester and could still get an A. And worse, if they then move to a class that challenges them again, then they are back to their old routine. This, of course, is what I'm worried about and I'm still trying to figure out if this happens or not.
As the department facilitator, I know that it is my job to get all of my department on board with challenging all students and getting them all engaged in the material, for all courses. With new teachers, I think this is easier to do than with older, more experienced ones. Especially today, the trend seems to be for teachers to be introduced to this "new" student-centered style. However, teachers that have been around for a while, as we know, are more likely to stick to their old habits and come up with many excuses/reasons to not change. I'm speaking in generalities at this point. My department members have seemed to be open to new styles and tools that can be used to effectively teach their curricula, but there may be some that just aren't sure how to do it. If there are teachers that focus on what is only in a textbook and need everything almost scripted to teach, how can that be changed?
I believe that in my department we need to be more consistent in how we teach. Obviously we all have our own individual styles for delivering information - I don't want to completely change everyone. However, getting back to my original question, I do want our courses to all have a certain level of rigor so that students feel challenged and interested in every course they take and they don't have to worry about what teacher they have next year. Maybe this is too idealistic, but it is a goal of mine. I don't want students to have to beg for one teacher because they are easy or because they are tough, or cross their fingers that its not so-and-so because of stories they've heard from their friends. I want consistency in the quality of education that will be provided to all of our math students. As a teacher, I don't want to be torn between filling in knowledge gaps for half of my students because they had one person and continuing at a certain pace because the other half had someone else.
As I write this, I realize that I've gone down a bunch of different roads that I didn't expect all because I want to know if I change my students in the way they problem solve for the better. It boils down to this: I want the comfort of knowing that what I do is making a difference or the frustration knowing that I need to change to make that happen. Right now I'm somewhere in the middle. I'm not exactly sure how to get an accurate answer to this question, which is quite disheartening. Ideally, I could have the same group of students two years in a row and see if what I've done has stuck with them. But even that wouldn't tell me if my strategies continue when they leave my room.
Of course, there is always the possibility that my students are just telling me what I want to hear, and they really do not like how I teach at all. Crap, that's a whole different set of issues...

## Thursday, February 7, 2013

### But Mr. Brandt (in a whiny voice)...

Today in geometry we had the "when am I ever going to use this" talk. It happens every year, in every section of every course, to every teacher, ever. It is my belief that the answer and approach to this question can make or break a class for some students. It was my moment to get them hooked or lose them.
I'm always honest with my students when it comes to this question: in reality, most of them will probably never solve these types of problems exactly as they appear in the world outside of high school and college. When I think back over what I learned in high school, maybe 15% is used on a regular basis? Maybe that estimate is off a bit and you think it should be higher or lower, I don't know. Even my college coursework, I use very little of it, and I teach this stuff. I think my students feel a sense of ease when I express this to them because I'm not sugar coating anything and coming up with ridiculous scenarios of how this can be used. In reality, I could come up with perfectly viable situations in where they might need the Pythagorean Theorem or Hero's Formula but they always have a response ready for me to counter act it. This doesn't bother me; I get it.
I do tell them that the real-world-ness is not the important part of education. The point of education is to learn for the sake of learning, and if that's not enough for them, the point of math is to be able to problem solve. I know that my students have not been challenged mentally before, and if they have it has not been overly strenuous or very often. I know that if I ask them to write me a paragraph on what it means to graph y=3x+5 they will give me a step-by-step procedure but little to no meaning (as I was explaining this to them most nodded their heads in agreement). I don't want step-by-step procedures from my students, and I don't want to teach that. I want them to see the connections within the mathematics. I want them to be able to problem solve. After they leave my class, I don't want them to ever look at a problem and give up without ever trying because 'it looks too hard.' My job, as an educator, is to show them that all of this math stuff can be used somewhere, and somewhere else, and somewhere else, and if you manipulate it a little bit, somewhere else. I want to give them challenging problems that they start, and struggle with, and from the solutions we can learn new things. On the last day of the semester, if I give them a super challenging problem, I want them to remember what we did in the first week of class and be able to recognize that it might work in this new scenario, even though we've never explicitly done it that way before. I want them to think critically, think through the problems, try new solutions, verify their work, and most of all not be afraid to do it.
So, to answer the question of 'when am I ever going to use this in my life,' I usually respond with the above explanation, attached with a "I don't know, and you don't either." My students don't know when/if they'll ever encounter any of this geometry stuff outside of school or in another class. But, if they do, I want them to be prepared and confident in their ability to use it. After all of this and possibly some more conversation, my students are on board. They get it. They've never had a teacher tell them they may never use this stuff, but they appreciate the honesty and can understand the logic behind my purpose.
Students like to be challenged, even if they complain through the entire process. When they see the solution at the end, or better yet when they get their on their own, there is a sense of pride. This whole 'real-world' argument that they present could be a defense mechanism to get out of doing the work, but more importantly it's because they have a desire to see some usefulness in the material. Students need relevance, but not necessarily always in a real-world context. If we as teachers can hook them, follow and encourage their thought process, and make them thirsty for more, we're on a path to success, and they are too.
I'm sure I'll have this conversation with my students again throughout the year, it always happens more than once, and I have no issue with that. I will do my best to keep them engaged and challenge their thinking, and most of all blow their minds, with math and all that it can do. If they leave my room at the end of the year and feel no different about math or are not any more confident in their problem solving ability than the day they walked in, then I have some serious reflecting to do. I think so far I've got them hooked. I believe they trust me to not lead them astray. That's the first step and now its time, in the words of a fellow colleague, to change lives.
I said a lot of things...

## Tuesday, January 1, 2013

### New Year, New Idea

I watched Mr. Cornally's TED talk, and immediately wished I worked in a school that operated the way he described. If you haven't watched it, please go see it before you do anything else. If he ever starts his own school, I want to work there. I shared it with some of my colleagues and even some of my students, and everyone responded with the same positive attitude saying it would be awesome to be able to get excited about education. The students that watched it expressed their love for studying something that they wanted to explore. This is what creates life-long learners and quality education. So, after much thought and reflection, I've decided that I'm going to start a little project similar to what Mr. Cornally described. It might be a lot of work, but it's going to be fun.
Some background info: In my school, we have what's called and 'iSpartan' block. Essentially it is about one hour of time where students are in their homerooms and they are expected to be working on assignments that teachers have provided. These assignments are meant to be ones that support and enhance current material and not just extra homework. During this time, students are also allowed to travel to their teachers to receive remediation if they have below a 70%. This block of time and policy is new for this year, and after watching it in action for one semester, I've noticed that many of my homeroom students either have not been assigned work or do not complete it in school (they wait until they get home or don't do it entirely). I do not like watching students do nothing during an hour of the school day when they could be using this time to their advantage. For my homeroom students, when they 'have nothing to do' during this time, I'm going to have them complete an independent research topic of their choosing. Below is an outline of what I've come up with so far. It is only a draft and needs some work, but it is what I've thought of off the top of my head. If you have any thoughts as to how this can be improved, please let me know. This is the first time I'm doing this and (as far as I know) the first time anyone in my district is pursuing anything similar.
----------------------------------------------------------------------------------------------------------------------

iSpartan Independent Project

Purpose: To allow students to explore any topic of their choosing in any way they desire. The bigger picture – to explore your creativity

Who: Homeroom students, Spartan tributes, select faculty as needed

Requirements: Do something interesting that exercises your creativity and pushes your limits, forcing you to dive deep into the material at hand.
Create a presentation to share with the class that accurately explains what you did with great detail

Procedure: Students will come up with a subject, project, topic, etc. that they are deeply and personally interested in and will share with me. Together, we will communicate on how to make it specific and engaging to develop a product that they can share with the class. Students will work on their own, with help from a faculty mentor as needed, throughout the course of the semester to create this end product. Some work may have to be done outside of class. For students who are having trouble coming up with an idea, I will work with them to discover their interests and from that develop a topic to explore.

Timeline: Projects must be completed and presented to class before the end of the year. Students will sign up for presentation dates as they arrive at the end of their project. When students are close to finishing, they will meet with me to ensure that they’ve got into great detail. Students may explore more than one topic if they wish depending on time.

Ideas     : English - Write poetry, short story, screenplay, Snap Judgement-type stories, film original movie, book report, research paper (avoid if possible), organize flash mob and explain purpose/explore effects that it had (http://improveverywhere.com/)
Science – build model rocket, Arduino electronics (http://www.projectallusion.com/1/post/2010/7/musical-handrail-using-the-mux-shield.html 2:00) (http://www.instructables.com/community/Piano-StairsFloor/), invent something, photography, robotics
Arts – drawings, paintings, etc. that share common theme, write music/album and explain meaning, learn to cook something awesome, build something awesome and huge out of legos, www.songreader.net
Math – physics, find math in something you enjoy, explore topic to its fullest, learn subject we don’t offer
History – research historical event, reenact historical event, interview someone, go to museum/fieldtrip and report on what learned, have class take part in reenactment, film documentary
Explore How Stuff Works, How Stuff Is Made, How To Do Anything
Organize Event

Resources: Snap Judgement, Radiolab, TED, www.popsci.com/diy