Mathematical Modelling and Adventures in Blogging

Back in February, I blogged about iTAGs, the meet-up groups that aim to help teachers "build community, develop as activists, and link social justice issues with classroom practice." I've joined two of them: "Social Justice Educators on the Path to Cultural Relevancy" and "Locally Relevant Mathematics with the Community Based Mathematics Project."

For the latter, I was invited to blog about one of our recent sessions, where we discussed practices for "modelling" in mathematics classrooms. I am re-posting my entry below, or you can view it in its original post on the Community Based Mathematics Project website.

[Note, since I'm a proofreader by trade and a bit of a perfectionist in my own work: I stuck with the double-L spelling of "modelling" rather than "modeling," even though I like the latter more, because the former was used more consistently in the literature we were drawing upon. However, whoever posted the entry used the single-L version in the title. Please forgive the inconsistency. And please forgive this ridiculous apology.]

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The Locally Relevant Mathematics iTAG is a place where we discuss ways to make mathematics lessons personally meaningful and culturally responsive -- but we also keep a focus on the way that these responsive practices can help promote particular forms of inquiry and mathematical understanding. For the latest session, we looked at modelling as a great classroom practice to support all of these goals.

But what actually is modelling?

Admittedly, it's not something I could clearly explain before, and even now it's still a bit tricky to define. As we discussed it, modelling is different from traditional problem-solving in that the "problem" is inextricably tied to the context, and students have to engage with a given scenario in order to figure out best how to approach it with mathematics.

Because that seems abstract and not very descriptive (it did to me!), we spent much of our meeting doing an example of a modelling task. We were given a scenario: given a particular set of Philadelphia neighborhoods, we were tasked with determining how many basketball courts would be an ideal number to build in each. Rather than provide us with a particular approach to take, or data to support us, our instructor instead asked us to brainstorm what kinds of data would be most important to help us make that decision. With a whiteboard full of potential data sources, we then discussed and narrowed down the list of desired data. Our instructor then let us know which data he had available for us. Working in small groups, we were free to request data and develop our own approaches and answers. Finally, we shared out with the class, explaining the various methods and responses we had developed.

While modelling tasks don't have to be this abstract, what mattered about this task was that there was no way for us to strip away the context and say, "oh, this is the formula the teacher wants us to use." We had to deal with the scenario as a real-world challenge, grapple with the data, and determine the approach that would most allow us to be successful.

While it might be hard to explain exactly what modelling is, it's easier to see how this kind of exercise fits the mission of our iTAG. First, the scenarios and contexts can be carefully chosen to be engagingly relevant to students' lives and experiences. Second, it uses that connection to challenge students to apply their understandings of mathematics in a way that can promote higher-level thinking and deeper understandings. This sort of task is more of a challenge to create, and it requires a lot of trust in one's students to provide them this sort of relatively open exploration; however, I remain convinced that the potential rewards make it entirely worthwhile.

After our modelling activity, one of the ITAG members shared valuable ideas from an article on "launching complex tasks" in the classroom. The article spelled out ways to get a complex problem started, like 1) Discuss key features of the context, 2) Discuss key mathematical ideas, 3) Use students' ideas to develop common language for the features of the problem, and 4) Avoid giving students a particular solution method. Using these strategies when launching a modelling problem, she said, made the process go smoother.

The article is "Launching Complex Tasks" by Kara Jackson, Emily Shahan, Lynsey Gibbons, and Paul Cobb. It was published in Mathematics Teaching in the Middle School (Volume 18, No. 1, August 2012).

Let's talk about silly things

Okay. Had a frustrating afternoon, and am feeling unmotivated by blogging at the moment. Some things I just really don't feel like writing about at this particular moment:

• Job hunting
• Philadelphia schools policy
• Things that I really need to get done right now (taxes, lesson planning, my final portfolio, etc.)
• The weather
• The Philadelphia Phillies

I'll get back to writing about some of those things soon. In the meantime, there's still plenty of things that can cheer me up. For instance, I just stumbled upon a keyboard shortcut that changed my font to red. Didn't know I could do that, doubt I'll ever do it again, but for the moment it's magical.

Some nerdy things can cheer me up too; for instance, earlier this month, I held my annual Pi(e) day celebration (naturally on March 14th - AKA 3.14), where a bunch of my friends, including a sizeable number from my TEP cohort, came together and baked a LOT of pies. I'm also enthusiastic at the moment about some events (okay, why is it doing it NOW???) about some events coming up, like Gloria Ladson-Billings coming to speak here next week, or trying out for this the following weekend. 

And then, of course, there's the thing that constantly lifts my spirits: the ridiculous, ridiculous, tiny human beings I'm surrounded by each day as I student teach.

Some of the strangest highlights:

• There's some questions you don't really want to answer as a Kindergarten teacher...so it's nice when kids have their own answers:
• Student A: (Reading a book about pandas) Mr. Gottschalk, what does "mating" mean?
• Me: ...
• A: Does it mean, making really good friends with someone?
• Me: ...
• Me: ...Kinda!

...And then there was only Term V

To recap, for those of you just joining us:

Although we're studying a lot of different things at once in the Teacher Education Program, our year is generally divided into five thematic terms.

• Term I was the summer. It focused on the context for teaching. We studied "School and Society," among other things, and our final product was a study of the neighborhood where we would be student teaching. 
• Term II was the first half of the fall. It focused on getting to know students as individual learners, as well as theoretical approaches to teaching and basic assessment & instructional strategies. Our Term II assignment was a study of a specific student from our classroom.
• Term III was the remainder of the fall. We focused on lesson planning, ultimately developing and teaching small-group lessons in each of the major subjects.
• Term IV is the beginning of the spring semester, and focuses on curriculum planning and working towards enduring understandings. Over the course of Term IV, we develop a full, cross-curricular lesson plan for a two-week full takeover.

And those are all now officially DONE! I'll write a recap about Term IV soon, but for now, I just wanted to report that I have finally reached...

• Term V: Praxis. Term V is different from the other terms -- we're only taking one class a week (one session of seminar, rather than two), but we're student teaching full-time now, ultimately building up to a two-week full-responsibility teaching takeover. There's not technically a "Term V assignment" -- but instead, we're working on finalizing our Master's Portfolios. We'll be going over more of the details tomorrow, but the TEP website describes this assignment as a "web-based portfolio addressing a critical issue that rises from their teaching and supporting their arguments with artifacts from their teaching experience." 

And that's really it. At this point in the year, "Classwork" has frankly subsided to Priority #3 -- behind "Student Teaching" and "Finding a Job," but still ahead of "Cleaning My Kitchen" and "Updating My Blog."

100th Day of School!

I don't recall the 100th day of school being a big deal. I vaguely remember one of my classrooms having a 180-day timeline around the top of the walls, but otherwise I don't have any memory of knowing, or caring about, what day it was. It was early in the year, and a lot of school remained. It was late in the year, and summer was approaching. I liked school a lot, but come on - summer vacation! The calendar was all about counting down, not counting up.

Well, I don't know if times have changed, if Philadelphia is different from Charlottesville, or what, but the 100th Day seems to be a big deal in a lot of schools here. I'd been hearing about the 100th day since the beginning of the school year - watching teachers counting on days, telling students excitedly that they would have a big celebration when they counted up to 100 - but I didn't really know what to expect.

Well, today was the 100th Day, and it was a Big Deal.

My classroom teacher went all out, as you can begin to see by the streamers, signs, and table decorations. At each student's seat was a $100 bill with their own face printed on it, and a colorful pencil. Students came dressed in 100-related clothing (one kid had a shirt that said in huge letters: "100th day!!! and Mom's birthday"). And yes, I got in on the action: