Making More Sense

     This is my first time teaching algebra 2. I have two goals: make sure my students master and enhance their algebra 1 skills, and have the students see/understand the material in a new way. So far, its early in the semester and we haven't gotten into anything too deep, but yesterday we started a familiar (but often over-complicated) topic, arithmetic sequences.
     "Given a sequence 4, 10, 16, 22, 28, ..., a) Find the next three terms, b) Find the 87th term, c) Find the sum of the first 26 terms." If you see this problem in a textbook, there is a good chance it is preceded by explicit and recursive formulas with some notation that often confuses students. Seeing an and an-1 is not the easiest thing to explain, nor is it overly easy for students to retain. It is not often used, certainly not consistently throughout math topics, and is usually only associated with this topic of sequences and series, geometric and arithmetic. When there is a great chance that the above is going to happen for my students, is there a way to teach this so that they will understand it and connect it to what we've previously done, or am I doomed to teach this tough notation and risk it getting in the way of their learning? Luckily I work with someone who thinks outside the box on nearly everything and has an incredible gift for showing students how patterns exist within math.
     We started the course by having students find linear functions with only a table of values. They are able to find the change in y and change in x, and from their develop the linear relationship. So instead of teaching arithmetic sequences in terms of explicit and recursive formulas, we related it to a skill our students already have.
     Why not put the sequence in a table of values and have our students come up with the 'explicit formula' themselves? Is an arithmetic sequence not linear, always? Since they can see there is a common difference, a rate of change if you will, why not have them evaluate sequences in the same way they evaluated linear functions? Instead of an notation we can simply redefine x and y in the context of the problem. For our purposes, x will represent the term number and y the term itself. Done. Arithmetic sequences have just been taught in 10-15 minutes with a full understanding from my students. What about the 'recursive formula' you say? Do you see that common difference (aka slope) you just came up with kids? Continue adding that and you have your recursive formula. No confusing an-1 notation, just sense-making within the material.

     I've never taught this topic before this semester, but teaching it this way has made immediate sense with my students. They like it, they get it, and they are good at it. When I see other students in our building that are learning it in a traditional textbook sense, they are confused and constantly asking questions as to what is going on. The notation is making them lost and keeping them from seeing what's going on. When I present it to them like this, light bulbs go off and they wonder why they didn't notice this relationship before. This has quickly become a topic a enjoy teaching because it is now something that is easily understandable. We haven't gotten to geometric sequences yet, but you can bet I'll be teaching it in a very similar matter, in the context of finding an exponential equation from a table. I expect the same kind of success when I get there, assuming I'm able to teach them the table portion of exponentials effectively.
     But what about the sum of n terms? Well, so far I've taught it just by developing the formula; showing that the first and last terms, second and second last terms, etc. all add to the same value and you have n/2 of those sums (with a little more exploration and detail that that). But, I could also follow a similar path as above and have my students generate a table with y being the cumulative totals for the terms x and below. I could teach my students how to find a quadratic equation from a table by using the second differences and they could go from there. I intend on making this connection once I start quadratics, but we're still in linear land. We'll get there.
     I'm all about teaching math in a way that makes sense to students, but doesn't sacrifice the meaning and understanding of the material. I don't want my students to get bogged down by notation and formulas. I want them to see math for what it truly is, a study of patterns. I want them to make connections to previous material that they've learned and see how it all connects. Very few textbooks do this, and I'm finding that I'm learning more ways to do this through talking to other math teachers and exploring the math myself from scratch (which I need to do more of).
     By the way, it also helped a great deal that I started teaching this topic by using Dan Meyer's triangle toothpick problem. I imagine providing a great context for the lesson helped tremendously with the engagement process from the students. I also continued and developed the concept with the help of Fawn Nguyen's visual patterns. A many thanks goes out to these great educators for creating and sharing these resources with the MTBoS.

#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.

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?

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.

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

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.

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?

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.

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...

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...

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. 

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. 

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. 

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...

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.