A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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My preservice elementary course this semester is an embedded field experience. Each week I write or find some lesson for the 3rd graders, and they teach in groups of 2 to 3 or 4 3rd graders, and then reteach in our next class period. Each class I sit in with a group, and everyone has some time to assess and reflect themselves and the learners while I debrief with that group. Before the first time teaching, at least, we try to rehearse together. As a whole, this is how I want to teach teacher prep from here on out. We're getting to less content, but I see so much more learning.
This week I had a lesson planned for Thursday, Pi Day, that had nothing to do with π. It was on these terrific Naoki Inaba place value puzzles that Jenna Laib shared. But with Pi Day approaching, and #MTBOS talk of activities and Scrooges, how could I help but think of a lesson?
Probably won't do a pi day activity with the 3rd graders, but if I did, I think it would be to make circles as best you can with lengths of string, then measure guess how far across.
Maybe better in reverse. You lay out string for a diameter, students cut a length to be the circle, and the winner is the best circle?
Yes, easier to start with the diameter and estimate and cut circumferences. Might be easier with thick paper strips, which have a stiffness and taped together will hold a roughly circular shape?
So I felt comfortable offering my PSTs a choice. Pi Day Lesson or the puzzle lesson. If they chose the puzzle lesson, we'd talk about reteaching it or getting back on a Tuesday first schedule. (So they have the weekend to prepare. That was the original idea, but we've had... weather.) To my surprise they overwhelmingly wanted a Pi Day lesson. Oh, crap!
For the objective, I definitely wanted learners to come away with the idea that π had to do with circles, to get to do some Visual -> Properties Van Hiele type experiences, and that it's a little surprising how far it is around a circle. Here's the lesson plan.
It starts with a quick activity that I think was first inspired by Patrick Honner, who had a lesson on a day like 11/11/11 where he had an equilateral triangle drawing contest. Who can draw the best ____ is a part of my repertoire now. It's just perfect how it goes from the visual to properties as people argue over which is best. Not without resistance. "Because it looks the most like a ____" Hmmm. So what makes a shape look like a _____?
Then the competition begins. A strip of paper cut out to be the diameter is placed, and learners try to cut out a circle that - when in it's most perfect circle shape - is the closest to having that diameter. They record the data of what they tried, and indicate which was closest. Then repeat for a longer length and then an even longer length. In the group I was watching, Sophie said at the 3rd one, "I think I want it about 3 times as long. Maybe a little longer." Why? "It's just a guess?" Did she connect with pi? Build from the data? Visual instinct? She wouldn't get more specific.
The teachers found that learners were engaged. They liked the making aspect. They definitely enjoyed the competition. A lot of good talk about circles, and a little bit about circumference. So, maybe file this lesson away for next year? 3.1420 is just around the corner.
I've been thinking a lot about mathematical modeling this semester. I'm teaching a preservice secondary class, and we're trying Catalyzing Change as a text. It mentions modeling 20 times in 100 pages, and definitely indicates it as one of the primary motivations for mathematics' importance.
"A mathematical model is a mathematical representation of a particular real-world process or phenomenon that is under examination, in an attempt to describe, explore, or understand it. When students engage in mathematical modeling, they often have the opportunity to leverage mathematics to understand and critique the world. Mathematical modeling is the creative, often collaborative, process of developing these representations. Modeling always requires decision making that involves determining which aspects of the phenomenon to include in the model and which to suppress or ignore and what kind of mathematical representation to use. As noted by the mathematician Henry Pollak (2012), throughout the modeling process, both the real-world situation and the mathematics must be taken seriously."
I rewrote my content standards for the course trying keep this in mind. I was recently getting observed for a personnel thing, and it was a day we were thinking about modeling. The observer was David Austin, who's teaching a new modeling course that we're offering as a part of developing an mathematics emphasis for the math major. I asked him to share the idea of the course and we tried to connect it to them to encourage future teachers to take it. Part of how we addressed that was a classic trigonometry problem of fitting a cosine model to the height of a seat on a ferris wheel. But they weren't using video of a real Ferris wheel. Why would we need a trig function for a real Ferris wheel? So I'm pretty sure that this was not real world modeling.
Then Dan Meyer took on modeling last week. He was at a panel with the folks who produced THE report about this, GAIMME. (People just say 'game'.) They identify four aspects to mathematical modeling. It's real world that bothers Dan. He said/writes:
If your definition of “real world” labels the US tax code as real and polygons as non-real, your definition is not useful. To most US K-12 students, the US tax code is very non-real and polygons are very real.
If you define “real-world” as a property that is binary rather than continuous, that is fixed across all cultures and time rather than relative and mutable, if your definition doesn’t account for the ways (per Freudenthal) that contexts becomereal in someone’s mind, it isn’t useful.
And if your distinction between “mathematical modeling” and “learning” depends on “real world,” a descriptor without a definition, it isn’t a meaningful distinction.
I love Dan's willingness to wade into a fight, and to constantly rethink and refine his ideas about teaching and learning. But I don't see how he's helping here. Real world may not have a sharp mathematical definition, but the idea that we can bring things from the real world into the math class is helpful. Many of Dan's first big ideas were about this, and I don't think he's denying it now. He also wrote a lot about pseudo context, which was helpful in getting at the idea that real world is not the be all end all.
I don't care about labeling or not labeling tasks as real world or not. I do care about the process the learners are engaged in with these problems.
GAIMME outlines this as the process:
Formulating the problem or question
Stating assumptions (often requiring simplifications of the real situation) and defining variables
Restating the problem or question mathematically
Solving the problem in the mathematical model
Analyzing and assessing the solution and the mathematical model • Refining the model, going back to the first steps if necessary
Reporting the results
Paula Beardell Krieg was sharing some of her usual awesome mathart, and it led to a quick conversation. She made an image that was interesting to her with trigonometric functions then did her arty magic to make it into paper that she then origamied into boxes. I commented that I'd been wondering if the modeling aspect was part of what I found so engaging about engaging in art in math class. (I was working on a David Mrugala idea, and, though not real world, I tried a lot of different functions trying to get different effects.) Getting the math to make what you want is messy, open and genuine. She said:
This idea of starting with something fuzzy then refining resonates with me in two ways. One, mathematically, is that more and more it seems like (from the glimpses I've had of higher level math) is that what mathematicians are doing is always trying to get more and more accurate models. ... What the artist does is similar in that we start with a sketch then keep refining until we decided we've gotten close enough. When I am playing with these functions to get images that I like, what's really awesome is that I get to use the math that I know as well as learn new, or refresh myself, with new stuff. It's an amazing process for me, as I have to push my math envelope to push my art envelope. It is so satisfying, but it does require so much mucking around. That's the mess that you talk about. It's also the play that I think about for my own work and for the work I do with kids. So much discovery in mucking about. I know you know that this is where so much learning happens. I am convinced that imposing too much structure (which is different than NO structure) sucks the potential out of learning. No structure can do the same. Finding that sweet spot, which is actually quite large, is a good place to aim for.
For Dan this leads into a big idea "Teachers need fewer ideas about teaching." Whoa! I'm definitely not ready to get into that. But I want one of those ideas for math teachers to be "learners need to be doing mathematics" and some of what doing math is is encapsulated in this process.
The end of the conversation (for now) with Paula was a direction for me to pursue these ideas.
Definitely we're on the same wavelength here. The advantage of art over modeling is that the learner does more of the problem posing. When I pose the problem or provide the data, maybe some are interested, others not. But art is them finding their own problem. The down side is that not everyone's interested in making art, and I don't know enough about selling that.
When I'm working with kids I find it's helpful not to think of what we're doing as making art or of having fun. What is more useful is to ask them to see what they can do with what I give them, and to try encourage a sense of play. The results I'm looking for has less to do with art and more to do with discovery. I did something like this with 2nd graders this past week. I may tweet about it ....
So it comes back to play! I think this is inherently tied up in the first step: the mathematician poses the problem.
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