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.

So Many Students (+two teachers)

     This week marked the beginning of my team teaching endeavor with the beginning of a new semester. A colleague of mine and I are teaching algebra 2 and pre-calculus together, combining our classes in a large group instruction room, and enjoying every minute of it. It's been an intense week, but a productive and overly positive one as well.
     The first day we started the students all together in the large room, giving them zero notice of what they were about to get into. According to their schedules, they all thought that their math classes would be as they were in the past, sitting in rows of desks listening to their teacher teach. They were massively confused, and some curious, when they were directed to a large room on the other side of the building. We explained to them what we were going to do and how the system was going to work. We could see some faces excited about this opportunity and others were nervous.
     Our algebra 2 class contains about 60 students, comprised of sophomores and juniors; it is a huge group. We didn't fully realize what 60 students looked like until that first day, and we got nervous ourselves. Our biggest fear is that a large number of students will get lost in the mix and we will be missing them and doing them a disservice. Sixty students is going to be tough to keep track of, but we're determined to come up with as many strategies as possible to do so. After the first week, we've already noticed that the overwhelming majority of these students are on board with this experiment of team teaching, which is great. With the clientele being younger, we were nervous they wouldn't want to change, but they're good to go. The good news is that after this first complete week, we've noticed that the students are looking out for each other. We have them sitting in tables of (at most) eight. Each table seems to have a few students that struggle, but also a few students that excel. Those excelling students have naturally been helping the others, but not in a "here's the answer" kind of way, more of a "let me explain what's going on and how to solve" kind of way. We did not prompt this in any way, it just kind of naturally happened. Students are asking us and their classmates questions and staying on top of the material. As we walk around, we hear the conversations going on at each table, and they are all about the topic at hand. There is a tremendous amount of support within this group, and it is truly incredible to watch.  We did our first true, in depth checkpoint after three days and the results were positive. Students are showing us that they know what is going on for the most part, and there are not any major gaps in knowledge at this point. Hopefully all of this continues and we're able to make great strides in the coming weeks.
     Our pre-calculus course has a slightly different feel to it. There are only about 40 students in this group, and they consist mainly of juniors with some seniors. We get the feeling that these students are not as enthusiastic about this experiment they are a part of. From some feedback that we have received, they would prefer to be back in a traditional classroom, just like the old days. While we want to do what is best for our students, we are encouraging them to give this a shot for a few weeks and see how it plays out. We don't want to take away from anyone's educational experience, but we realize that their resistance could possibly be from a fear of change. We'll see how it plays out and seek their input constantly, but hopefully they jump on board and we can continue combining our classes.
     One observation that we have made already with these upper classmen is that they want to be challenged. We started on the first day with finding a linear equation, given a table of points. We tried to explain some things in a new light, but the students didn't bite. They weren't interested and they were not engaged. My colleague and I are not fans of reviewing algebra 1 topics at the beginning of courses, but we were going to build on this skill and wanted something to relate to. The second day we went into finding the quadratic function given a table of values, and the students were hooked. While the topic may not seem exciting, they have never done this before without a calculator 'quadratic regression' function. Doing it by hand was a foreign concept to them (and a challenge), but they saw the connection to the first day's material, and quickly latched on. We had every student's attention now that they were being challenged with the material, and we had much more participation. There was a huge difference between the first and second day in the attitudes of the students. We knew that pushing them was the key and we plan on doing it for the remainder of the semester. It made us think about how students feel in other classes, where they spend a week or two doing pure algebra 1 review. It's no wonder that students don't like math when they are put in those environments. If they are forced to sit through material that they have seen over and over again, and the know it, what is going to get them to enjoy the class from the start?  I have talked to other math teachers that feel they need to start every course (regardless of the students and the level) with a few days/weeks of algebra 1 review; they do not agree with our process. In my opinion, from the research that I've done, it is much more effective to dive into new material and review those previous concepts as necessary in the context of the new material. Why should I take two or three days to review solving linear equations when I can do it for a few minutes (only if I need to) when we need that concept to master another? But, that's a different conversation. Bottom line: our students want to learn and they want a challenge - we're going to give it to them.
     We remain excited about this opportunity of combining our classes and are determined to do everything we can to make it a success. There will be bumps along the road, there will be times when we lose students, and there will be adjustments made constantly. I don't think that we'll ever have it down to a science with a teaching formula that works perfectly, but with the trust of our students, I think we can come pretty darn close. As the semester progresses, I will try to post updates on some of the procedures we come up with, as well as the observations we make/feedback we get from students. Wish us luck, and if you ever get the opportunity to do something like this, I highly recommend you give it a shot.

#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?

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!

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