## Friday, February 21, 2014

### 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.

## Saturday, January 25, 2014

### 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.