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.

Thoughts On Student Directed Curriculum?

WARNING: Lengthy, lack of visuals/humor. I'm going to hit you with some knowledge here (and I hope you, in turn, smack me with some feedback as well).

This is my sixth year teaching geometry and I'm pretty much at that point where I can walk in to my classes ask them what we talked about yesterday and I can go to town. I've always wanted to be at this point in my career, where I can spend time focusing on increasing the quality of questions and researching new ways of teaching rather than taking a tremendous amount of time just figuring out what I'm doing the next day, creating problems, etc. Because I have most of the ground work laid out, I've been able to try some really cool (and some not-so-cool) things over the last few years. For some of my units I've gotten rid of tests and created projects, for some I've made them more discovery-based and student-directed, and for some lessons I've created some really involved questions that require some incredible thinking from my students to solve. This year an idea I've been toying with is the student-directed curriculum. Its got some pros and cons to it and I'm not quite sure what I'm going to do.

Our geometry curriculum used to be like almost every other math class: follow the textbook. Unfortunately, we have UCSMP which is terrible in my opinion. When I rewrote the curriculum two years ago I had one goal in mind: organize it in a way that will make sense to the students and where connections can be made. I got tired to teaching topics and having to jump all over the place to struggle to draw the lines between everything. The beauty of geometry is that is all related and our students were not seeing that. Now, even though technically we jump all over the book (although no one in our dept uses the book anymore) I've noticed student achievement go up and we've been able to create deeper questions. The students don't care that they don't have a book to follow because the way that its organized makes sense to them. They would rather reference pg. 18 and pg. 262 in the same day than have everything disorganized and all over the place.

This semester I took the first three units (basic vocab, quadrilaterals, triangles) and mashed them together as opposed to teaching them separately. Now these units include things like all of the types of angles, symmetry, trig., and more, so they're pretty heavy on material, but I wanted to have it make even more sense for my students (if that's possible). As I thought about the curriculum, I realized that it could be organized in a number of different ways and still emphasize all of the interconnectedness of the world of geometry, but how could I have it be the absolute best for my classes? Or for anyone's classes for that matter?

 I started the year by asking my students what came to their minds when they thought about geometry. Their response: "Shapes and stuff." Me: "Name some shapes." Them: mur mur mumble mumble shapes blah blah ...and somehow I took it from there. I took the shapes they gave me and we started to dissect each one individually, exploring all the properties until there was nothing left to talk about, making connections as we went. I never had to say "Ok, we're done with that figure. Let's move onto the next one." because the properties naturally lead to more figures. Through student questioning I was able to completely cover the curriculum, plus some more.

Because this was my first time trying this, I limited myself to just the first three units as opposed to attacking the entire course this way. I'm teaching two sections of geometry and my hope was that I would be teaching them different concepts because their questioning would lead them to different places, but they'd end up at the same place in the end. I'll be honest, it got confusing. I no longer could stand in front of them and say "What did we do yesterday?" because I couldn't remember everything we'd discussed. I found myself keeping an unnaturally large number of post-it notes on my desk reminding me of what I've discussed with each class. I'd imagine when my third block asked my second block what to expect in class, they were surprised when they did something completely different.

I noticed that teaching this way caused achievement to increase compared to other years. Now, obviously I could just have an awesome batch of students (which I do), but I believe the way I taught had some impact as well. All of this has led me to ask myself, what if I taught the entire course this way?

Teaching an entire course through student questioning: innovative or something I should've been doing all along? Either way, there are some definite benefits and challenges to this approach. I believe my students would see some great success both in learning the basic knowledge and developing some higher order thinking skills. Teaching this way allows me to easily pull all kinds of topics together and potentially create some really cool projects. It would allow my students to really understand that math is all connected; its not separated into Alg 1, Alg 2, Geometry, Pre Calc, etc. but they're all based upon each other. It would also keep my students involved in the course. They would have complete ownership over the material because they would be determining what happens next. This also reinforces my philosophy of 'teach what makes sense, not what comes next in the book.' Looking through the PACCSS, I think I could hit everything with this style.

The tough part is that I have to almost be fully prepared to teach the entire course at any moment. Because I won't know where my students will lead me, I need to have everything ready to go on the first day. I'm sure I could predict a little bit as to where they would head, but I wouldn't know what they're going to do on a daily basis. If I taught this way I wouldn't want to push them in any direction unless they stall; I want them to be pushing me. Another challenge is keeping straight what I've covered and what I haven't in each class. If I would do this next semester, I teach three sections of geometry, that would be three different places in the curriculum simultaneously. I would be very fearful that I would forget to cover something or I'd start going over something in May that we discussed in February. While ideally this would be 100% student run, obviously there would have to be some questioning on my part to push them, which would give me some influence as to what's being discussed. My post-it note system of organization would fail rather quickly and I'd have to come up with something more efficient. I'd also need to create a new system for catching students up when they are absent. Maybe designate someone to constantly take picture of the board and post them online? There are some details to figure out.

As of now, I'm leaning towards doing this for one class instead of all three as a trial. This might help me get some of the details worked out before I push through entirely (of course, we all know what will happen if I do this: this will be the last year I teach geometry and I'll be back to the drawing board with new courses next year). It would also be fun to switch things up a little bit. I'm excited, and scared out of my mind, to try this. I'm not usually the most organized person in the world so this plan has potential to fall apart in a hurry. I'm hoping my desire outweighs any negatives that would potentially come out of this. The positives definitely outweigh the potential bumps in the road, and because of that I keep coming back to "I'd be an idiot not to do this!" I can't help but think of a quote from Mr. Pershan that I recently saw on Twitter (@mpershan), "If I'm not working really hard - if it isn't mentally exhausting, then I'm probably not getting better."

However, this style does kind of go against the 'common unit assessment' plan that my district has implemented for this year. Having common courses among teachers gets thrown out the window with this idea. Oh well... I gotta do what's best for the kids.

Do you, Mr. or Mrs. Reader, have any thoughts on this plan? Any positives or negatives that I didn't mention? Any ideas on how to overcome the negatives?

SDC (day 5)

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

Student Directed Curriculum (Day 2)

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

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

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

Student Directed Curriculum (kind of)

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

With A 'Lil Bit O' Algebra

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

iAwesome

So I just found out about Apple's iBooks Author app, and got extremely excited. The idea that I could create my own textbook fascinated me. I could be sure that my students received the information they needed and it corresponded perfectly with the way it was being taught. No information would be switched around, there wouldn't be any jumping around, and I wouldn't have to worry about teaching topics that aren't in the book. Even if I just made a textbook for my classroom, there could be tremendous benefits. By making 'personalized' text books we could cater to the needs of our students, our district, and our teaching styles. To me, this is much better than an online course (even though they are closely related) because the material can be written in the way that works for the course and the instructor. Earlier in education, curriculums were driven by textbooks. Teachers would follow a book from chapter 1 to chapter 20, one after another, and use all of the resources that came with that book. Now, with all of the common core and assessment anchors and standardized testing and everything else that's (invalid) and driving education, teachers are creating their own curriculums and moving all over the place in textbooks. If we need to create our own courses, it only makes sense to create our own books that follow that course.
Now, I'm not 100% sold on the idea of an e-textbook (and my school does not have classroom sets of iPads), but still, just the thought of it is awesome. It would be something that I would love to try.
My only quarrel: the software only runs on a mac, which I do not have. I need a PC version so I can create a textbook (or $$$ so I can buy a mac and iPad).