Thursday, February 21, 2013

Ugh, What Now?

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

Wednesday, February 13, 2013

Am I Really Making A Difference?

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

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

Thursday, February 7, 2013

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

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