A new trimester is upon us in St. Francis, MN which means a new group of advanced algebra students as well as two classes full of squirrely 9th graders.  I’m amazed that these sets of students can have things in common and a lesson for one class can serve as a bell-ringer for another.  I have said in the past that my favorite activities are the ones that can be used across multiple ability levels and this task is no exception.

This week, in advanced algebra, we’ve been working on problems that allow the students to connect specific patterns and examples to general formulae.  I feel that this trimester, I have done a much better job of sequencing the class problems in a way that has help build student confidence in the problem solving process.  As I’ve done in the past, I chose some nrich problems that have a low barrier to entry and a high ceiling.  These problems feel like number play:  Pair Products, Always a Multiple, Think of Two Numbers, and Calendar Capers.  Although I’ve had the occasional moan from students who prefer their math to be in lecture/practice format, I’ve seen much more willingness to engage in the problem-solving process this time around.

One particularly memorable day, we used a Math Forum problem called Baffling Brother in which a brother is attempting to amaze his younger sister by having her choose a number, perform some operations on the number and then telling her the result.  I’m disappointed that I didn’t think at the time to have the students act out this scenario.  That could have been spectacular!

These being upper level students, I always encourage them to attempt the “extra” for these problems.  On this task, they needed to come up with a number puzzle of their own that resulted in an answer of 7 each time.  I told them that I would be giving these number puzzles to my 9th grade classes to amaze.

Here are some examples:

What happened next I could not have predicted and was not an iota shy of completely awesome!  I presented one of these number puzzles to my 9th grade class and stood in the back of the room as I read the steps to them.  When they arrived at their final answers, I had them compare with one another.  I wish I had a camera on the room to capture the amazed look on their faces when they realized they all got an answer of 7.  Icing on the cake:  the advanced algebra students were very satisfied that they were able to amaze 9th graders with problems that they created.  I’ll call that one a win for engaging kids in “boring” old, non-applicable, relevant Algebra.

The trimester schedule that our school uses has many benefits, including 68 minute periods.  This seems to be the perfect amount of time for me as a math teacher to give the right amount of weight to each part of a lesson. One major drawback, however, is that sometimes I’m not ready to let them go.  Maybe it is an overall resistance to change, but more likely I feel that after 13 weeks, my classes are starting to make some major progress down the persistent problem solving path.  And then I have to let them go.  There are very few students that end up in my class more than once per trimester since the majority of what I teach is College Algebra and Probability and Statistics (both one trimester courses).

In my college algebra course, I’ve seen the progression and improvement in their problem solving abilities, but I wanted to see what they would say if I asked how they thought they had progressed.

So, as the last question on their final exam, I asked them what skills developed in this course they felt were most valuable.  (Oddly enough, none of them said factoring a quadratic equation.) But they did say some things that helped solidify my approach to this class.  Most frequently, students mentioned that they have improved their problem solving skills.  I’m confident that they are referencing problem solving skills in relation to rich problems, since that is a majority of what we did in this class.  Another common comment had to do with multiple approaches to solving problems.  Many students mentioned that they didn’t realize how many different ways there are to approach a problem until these varieties were laid out by other students. I, too, improved my ability to look for and appreciate the diversity in problem solving strategies.

My favorite comment overall was from a girl who works very hard but who hesitates to share her ideas with the class.  She states, “I can problem solve without a set procedure and now I feel like I can solve anything.”  Bingo.  I hope that I can help build that confidence as I work to improve some of the methods I used in this next trimester.

Part of my goal for my students this year is to help them become comfortable being mathematically curious.  In an effort to help students develop a growth mindset and to facilitate learning opportunities that foster this, I try to pose questions that allow time for exploration.  This has been more difficult than I thought it would be, since I am somewhat attempting to undo 10-11 years of an unwritten didactic contract:  teachers instruct, students passively absorb and regurgitate information.   Repeat as many times as necessary.

Most of the trimester I have gotten questions like, “Is this what you are looking for?” or “Are we going to be tested on this?” I thought that as the trimester progressed, this would happen less often.  I was wrong.  I had to admit to myself that as long as I was in charge of giving these kids a grade and as long as their grades remained a driving force in their outlook on education, I didn’t see this changing much.

I wanted to make note, however, of the progress these kids have made toward being more mathematically curious.  I’ve exposed them to some interesting graphs, and some students have shared how they expanded those ideas to make even more intricate versions of those graphs.   (Ever wondered what y=xsin(1/x^13) looks like?)

I had another student explore the graph of y = 1/(x-2) + 1/(x-2) and wonder if there was an equation we could write that would isolate just the middle portion, between the two vertical asymptotes.  He thought that it looked cubic, so he played around with a number of cubic functions, but couldn’t get the graph to fit quite right.  I commended him for his efforts, because this was the type of student known for wanting his math straight to the point.  I asked him the range of a cubic function compared to the range of the graph he was trying to match, and he quickly saw that his initial thoughts on a cubic function were incorrect.  I challenged him to keep searching for an equation (or two) that would match the portion of the graph he wished to isolate.  This was an important moment for both me and this student.  I had gotten him to explore and wonder with something that had no external purpose.  He did this for the meer wonderment of whether it would work or not.

This was fantastic and worth reflecting over for me as an algebra teacher.  Much of high school algebra is taught in a dry, procedural manner.  Unfortunately, the kids expect it this way, and the high achieving kids even want it this way.  They’ve been successful with it so far, why change it?  I hope as I continue to pose mathematical questions to these kids that they continue to push their understanding forward by exploring.

Kate Nowak was the one, whether she knows it or not, that gave me the convincing boost I needed to start blogging.  So I figured I owed it to her to respond to her request of “why I blog.”

1. What hooked you on reading the blogs? Was it a particular post or person? Was it an initiative by the nice MTBoS folks? A colleague in your building got you into it? Desperation?

I was taking a PD class called Thinking Mathematics through my districts Teacher Academy.  As part of the course, we were to read a chapter from Accessible Mathematics.  Of course, I bought the whole book instantly because it was exactly what I needed to get me excited about changing some of my teaching practices to become a better teacher. The book was so easy to read, easy to follow and made so much sense.

Anyway, as part of this class, we were given time to develop a unit plan and formulate lessons.  I was determined to scour the internet for some good resources.  I found myself flooded with them.  I can’t recall the very first blog I came across, but I found that I needed to start reading blogs regularly because there were some great math teacher bloggers with some great ideas and who were open and freely willing to share their resources.  I was immediately hooked.  Good thing I already had an iphone.
2. What keeps you coming back? What’s the biggest thing you get out of reading and/or commenting?

I found that the more you give, the more you get.  I started blogging and commenting on blogs at about the same time.  The more I commented, the more I wanted to write more blog posts, the more I wanted to comment, the more I wanted to blog, the more I …you get the idea.

I am find that I always love to hear other people’s ideas face to face.  I knew that reading about other people’s ideas could be even more fun!

Just as students learn more about mathematics by talking to one another about mathematics,  we as teachers should take that same advice.  The more we collaborate across the web, the more multi-faceted our lessons can be.
3. If you write, why do you write? What’s the biggest thing you get out of it?

Enter Kate Nowak.  Once at a Global Math Department meeting, she mentioned something about why SHE started blogging.  She said she started blogging for herself, to get her teaching ideas out of her brain and to reflect on her lessons.  I took this to heart because I realized that if I was going to blog, my goal should be for self-reflection.

The first blog post I wrote was for Dan Meyer’s Makeover Monday.  It was the last week where he kills it with the Desmos Penny Circle.  I was very intrigued that there were other teachers across the country that cared about my input on a particular task.  It shouldn’t have been a surprise to me.  I care what other teachers have to say, why shouldn’t other teachers care what I think?
4. If you chose to enter a room where I was going to talk about blogging for an hour (or however long you could stand it), what would you hope to be hearing from me? MTBoS cheerleading and/or tourism? How-to’s? Stories?

Your story.  How blogging transformed your teaching and your view of how teachers connect.  And how easy it is to get started.  I’d love to hear it.  Good luck, Kate.