Week 1: Why My First Day Activity Didn't Go At All As I Had Hoped (and Why That's Awesome)

Phew, the first week has come and gone and I found myself utterly exhausted on Friday. Thank goodness for the holiday weekend; I've been able to get more sleep in the past 2-3 days than I have in quite a while.

As part of the kickoff to our school year, I had my seniors work on an "opening day" activity that I lovingly borrowed/blatantly stole from Nadji (who blogs at Physix Coolisms!) that involves grids, writing your name (a lot), and using that data to generalize a pattern.

The activity I snagged is called "What Is Math?" and is described by Nadji from 4:25 to 12:30 of this First Day of School Activities presentation from Global Math Dept. I won't re-post the entire activity here, but basically the aim of the activity is to challenge students' perceptions of what it means to do math.


The Activity
The first part of the activity has students answer the following questions:

1. What is math? What does it mean to you?
2. List 7 mathematical words or phrases that come to mind when doing math.

Often, students respond to these questions with the mindset that "doing math" means working with numbers and calculations and equations.

After answering these questions, students then fill out several square grids by writing the letters of their name over and over again.

After doing that, the students shade the first letter of their first name and then fill out a table to record the patterns that show up.

From there, students are asked to make predictions about the patterns, such as:

• Predict what pattern would appear in a 41x41 grid.
• Predict how the patterns would be affected if the second letter of each name was shaded instead.
• Predict how the patterns would be affected if students started by writing their name in the bottom right corner and filling out the grid backwards.

And so on.

At the end of the activity, students are asked the beginning two questions again; by this point, the hope is that students will start to see that math is much more than just working with numbers and calculations and equations. There is much more to mathematics: finding patterns, making generalizations, predicting unknown events, thinking critically, etc.


How Things Went:
Before I get into this, one side (yes, side, not snide) comment: I had my students fill out grids up to 10x10. If I do this activity again, I might have them go up to 12x12. I have many students with names that are 6, 7, or 8 letters long, and their patterns don't really start to become apparent until the grids get bigger. As I checked in on students and looked through the tables they were filling out, it seemed to me that the "pattern of the patterns" would be more apparent if they had more data. Something to think about for next time.

At any rate, student responses to the opening two questions went pretty much as I expected. Many students came up with responses like "math is the study of numbers," or "math is the tool of Satan," and so forth. The lists of 7 mathematical terms often included "addition, subtraction, multiplication, division, square root, equation, numbers," and the like.

I decided to collect answers to the first two questions via Socrative, so I could quickly generate a bunch of text and then dump them into a Wordle. I thought it would be cool to generate a visual snapshot of student responses from before and after the activity so I could compare.

Here is the "before" Wordle:

As you can see, there's a great deal of "number-ish, calculation-y" stuff. I expected to see this.

Based on what I was seeing from the students as they were working on the activity and the conversations they were having (with each other and with me), I expected to see a dramatically different Wordle from the post-activity responses. After all, they were noticing patterns, making predictions about how patterns would look in grids that were far larger than they had time to fill out, and working together to describe a "rule" for making such a prediction. They weren't really doing "stuff with numbers."

So, here's how the post-activity Wordle turned out:

So uh... um... not really all that different. I mean, "patterns" showed up a lot more in this one, but there was still a bunch of "number-ish, calculation-y" stuff.

I'll admit, at first I was a little bummed that I seemingly hadn't changed very many minds or shifted very many paradigms after doing this activity.

But then I thought about it. And I became okay with it.

In fact, it's actually pretty awesome that I didn't change their minds so easily, and here's why:

This becomes a new challenge for me. This allows me to set a goal for myself. I want my students, by the end of the year, to understand that there's a lot more to mathematics than just crunching numbers and solving numerical problems.

Math is recognizing patterns and trends. Math is making use of those recognitions to make predictions. Math is critical thinking.

Math is art. Math is visual, spatial, tangible.

Math is freaking everywhere and freaking awesome.

I don't get to spend just one day trying to convince my students of this. I get to spend an entire year trying to convince my students how super-cool math is. I have a lot of convincing to do, but that's okay with me. I want to earn it.

That's one thing I learned from doing this activity. That's one thing this activity has given to me: a theme for this year: Math is freaking everywhere and freaking awesome.

It's going to be a great year.

Reflections From #precalcchat: Pre-Calculus Sequencing

I love Twitter chats with other teachers. It's a great way to make connections. It's a great way to get insight, ideas, and resources. It's also a fantastic opportunity to reflect on your own practice and to improve what you're doing in the classroom.

The Global Math Department hosts several weekly Twitter chats for math teachers on a variety of topics. Since I teach Pre-Calculus, I dropped in on the first #precalcchat of the school year last week; thanks to Mimi (I Hope This Old Train Breaks Down...) and Taoufik Nadji for hosting. Couldn't spend much time, but the topic of conversation captured my interest:

I loved that thought. It made me stop and think about how I sequence my Pre-Calculus course and why.

I start with Graphing and Functions first. To me, it's important for students to understand the basics of interpreting graphs of functions and becoming fluent with moving between different representations of a function (graphs, tables, equations). I find this to be a particularly vital theme that I want to drive home with my students, especially those who will be going on to AP Calculus or Calculus I/II in college.

Next, I follow a pretty standard sequence of Quadratics/Polynomials, Rational Functions, Exponential Functions, Logarithmic Functions, Trigonometric Functions, and Analytical Trigonometry. Again, I focus on these topics in particular to prepare my students for success in an AP Calculus course. Other topics such as Analytical Geometry, Series & Sequences, Polar Systems of Coordinates, Conics, etc. come afterward as time allows.

The other chatters all had brilliant things to say, so naturally I felt like I'm probably doing everything wrong (or maybe just some things wrong, and other things not-as-wrong).

When discussing how Pre-Calculus can seem like a re-teaching of Algebra II to students, Tina C (Drawing On Math) mentioned that her school starts with Trigonometry for that exact reason.

This was an interesting idea to quite a few of us: do Trig first semester, slowly build up conceptual understanding of the unit circle, graphing, transformations, identities, etc. Then, move into the other different functions second semester.

The more I think about doing Trig first, the more appealing it seems to me. I've always found that I never seem to have enough time to really properly teach Trig and I need to either rush a few things or cut some other stuff out. I think I probably always had the notion that Trig is "more difficult," and somehow it made sense to put the "harder stuff" at the end of the year. (That's excellent reasoning, isn't it?)

But really though, Trig is a bit of a stand-alone topic. It could go anywhere in the course sequence. There are certainly some underlying concepts that can be applied to other functions: graphing, transformations, moving fluently between representations, and so on. I usually think of these concepts as having to be taught and mastered before doing Trig, as if Trig is the "CHALLENGE MODE" of working with functions in Pre-Calculus.

Who's to say we can't use Trig to teach these concepts instead? Maybe my students would have greater success with Trig if I did it at the beginning of the year, built the concepts slowly with appropriate scaffolding, while still equipping students to be successful in working with other functions. I may have to try it out one of these years. (I already have this year mapped out -- maybe next year?)

Anyway, some great food for thought.

I'm looking forward to more of these chats this school year, and hopefully I'll find time to continue blogging & reflecting on what I take away from them.