Tina Cardone typically blogs at Drawing on Math. She attended the Northwestern Math Conference earlier this month and has recapped her first day and second two days there, including sessions on algebra, algebraic thinking, modeling, problem solving, and engagement.
#NCTMNOLA is almost on us and it remains to be seen how much recapping will happen. I’ve posted an excerpt of my own schedule on my personal blog and I intend to recap a lot of it. I have conflicts with great sessions in every slot, though, so it’d be great to get some of you to volunteer to recap even one session for this site.
Leave your info in the volunteer page if you’d like to help out and I’ll send along some brief instructions. Thanks, everybody, and safe travels to NOLA.
Avery wins the prize for Best Session Description by sneaking in the totally droll line, “All hail CCSSM MP3.”
He briefly dinged two-column proofs, the Disaster Island where we usually sequester conjecture and argument. I’m all for broading and deepening the definition of proof but I think Avery stretched it too far to include skills like estimation and asking us to justify our answers to the Locker Problem. Is “justify your answer” any different than “prove your conjecture”?
Later, he got into territory I found challenging and really well-thought-out. He showed how simple puzzles like Shikaku could be used to introduce the difference between axioms and theorems. He showed a Shikaku puzzle and its answer (below) and asked us, “What are the rules here?”
“The numbers define the area of a rectangle” and “the side lengths of those rectangles are integers” are axioms, without which the game wouldn’t make any sense. Theorems are the consequences of the axioms, like “Prime-numbered areas result in long, skinny rectangles with side-length 1.”
He also used a variation on the old game Mastermind to create a class-wide context for conjecturing, contradicting, and proving. Great stuff. Wish you were here.
Five interesting investigations with polygons [pdf]. Michael spent ten minutes prefacing the set, then let us investigate them for twenty minutes, and then asked a volunteer to debrief each one at the end.
If nothing else, it was a nice morning moment to talk about math with Internet friends. That was enough. But I’ve been struck also by how hard it is to make a given math concept more challenging for students and more interesting at the same time. We use bigger numbers. We mix in fractions and decimals. We lengthen the problem set. We time it.
For instance, once students understand how to find the sum of the interior angles of a polygon, it’s like, what do you do to make this more challenging and more interesting?
Michael introduced donut polygons:
Finding the interior angle sum of a donut polygon makes the original task more challenging and more interesting at the same time. In particular, it has a great stinger at the end when you find out whether or not a triangle inside of a pentagon has the same angle sum as a pentagon inside of a triangle.
Michael had two questions at the end that asked, basically, “Do your conclusions hold if there’s a dent in the polygon?” Then, “What about two dents in the polygon?” This messed me up a little bit, because, no, it shouldn’t matter, but then why would Serra include the two questions? Basically, Serra had your correspondent feeling briefly but completely off balance.
Great premise for a session:
Pose two lesson objectives. For instance:
- Students will be able to understand why the angles in a triangle always add to 180 degrees.
- Students will be able to understand how to calculate the shortest distance from a point to a line and then back to another point.
Allan then brought any resource you’d want, from low-tech to high-tech, everything from tracing paper and scissors to a class set of TI-Nspire’s. We used what we wanted to explore those objectives and then debated the merits of the analog and digital technologies.
The debate hit a high register pretty quickly. For the 180 degrees question, people tended to favor the diverse ways you could demonstrate it with paper. (Cutting, tracing, drawing, etc.) With the NSpire, you had one. You downloaded an applet from Allan and moved vertices around, watching the angle sum stay constant.
For the shortest path problem, most people preferred the calculator because you were able to set up the constraints and then drag a point around to see where the distance bottomed out. The only analog method we discussed was to draw the scenario and then use string to measure the different possibilities, slowly narrowing in on the answer.
For my part, I was bothered that we never discussed a) any other digital technology, aside from Texas Instruments calculators, or b) the cost of a class set of any kind of technology.
Granted, I probably make sport of Texas Instruments too much (and I’m hardly unbiased here) but truly I just find the user experience miserable. From the untouchable, low-DPI screens, to the time it takes me to find the right button out of the millions, to the mindless, button-pushing worksheets teachers have to pass out just to make the devices comprehensible, to the lurching way the cursor moves across the screen in Cabri, I find the whole experience pretty painful.
It would have been great to see the same problems approached with Geogebra or Desmos, for instance. Or even an Excel spreadsheet.
Then there’s the cost. All other things held equal, the high-tech solution will still cost thousands of dollars more per classroom. So we shouldn’t be talking about which solution comes out barely ahead of the other. Technology should shoulder the greater burden of proof here.
Bree claimed that stories are a useful medium for student learning. She backed this up with lots of citations that I hope she’ll share somewhere. [Update: She has.] She posed ideas for filtering our own lessons through the logic of stories – setting context, adding conflicts, etc. She closed by asking us try to create story-based lessons for exponentials and fraction division.
My thoughts went to St. Matthew Island, which I’ll link without elaboration.
Robert showed us this image and asked us to figure out how much it cost.
I’ve seen his lesson plan before but it didn’t prepare me for how interesting the math became.
We used the In-N-Out prices for a hamburger, a cheeseburger (a hamburger + cheese), and a Double-Double (a cheeseburger twice over) to figure out the cost of the 100×100 burger.
Our calculation was exactly right but we arrived at it differently. I used a system of three linear equations. My seatmates used a bit more conceptual creativity and got the same answer with a lot less computation. Robert highlighted all of these methods.
My takeaway: it’s really, really hard to describe in a text-based lesson plan all the awesome, heady moments it may provoke. I knew the lesson would be fun and productive. I didn’t know it would be this fun and productive. How do we create lesson plans that convey all those highs, rather than just the nuts and bolts of their implementation?
I think Christopher Danielson said most of what I was thinking during this somewhat odd father-daughter session. It’s difficult to describe the vibe that was in the room, with the presenters casually and sometimes clumsily taking turns describing then showing their videos. Near the end Vi grabbed a guitar for a rather brave musical performance that filled me with some kind of vicarious embarrassment, as if Fiona Apple had gone on stage thinking she was singing for lovelorn teens when in fact it was just those teens’ math teachers. Then again, I feel embarrassed for others quite easily.
Perhaps I shouldn’t be too critical. Some of the videos were pretty cool and who among us hasn’t had at least one “Hey guys, check out this thing on YouTube” kind of moment?
I was one of many who turned out for Dan Meyer’s session, which focused on why teachers should be using technology to capture, share, and resolve perplexity. What’s perplexity? Dan described it as “not confusion” but a “wanting to know, thinking you’re able to know.” Dan is careful to differentiate perplexity from engagement, as we’ve all been engaged in something that was simply tedious or boring.
This kind of perplexity describes a particular state of mind, one with more promise than the traditional definitions that describe perplexity as full of uncertainty and difficulty. However, when Dan speaks of “capturing” and “sharing” complexity, he’s not so much describing a state of mind as he’s describing the kinds of phenomena that provoke the asking of mathematical questions accompanied by an eagerness to mathematize. I’m hoping as Dan and others go forward we develop some sort of theoretical basis for these phenomena, or at very least, a useful classification system that can aid in task design. For example, I see the phenomena of scale frequently in Dan’s work, provoking questions like “How much might the big blue bear weigh if it were a real, live bear?”
For capturing perplexity, Dan showed various tools for finding and saving things from the web and the world. These tools included an RSS reader for following blogs and news sites, a tool for downloading and saving YouTube videos, a note-taking application, a tool for capturing audio memos, and the camera on your phone. The particular tools here don’t matter as much as knowing why to use them — they key is finding the tools that work well for you.
For sharing perplexity, Dan included technology like a computer with speakers and a projector, a document camera for showing student work, slideshow software, editors for photos and video, and a personal blog to “share the best stuff you do publicly.”
For resolving perplexity, Dan made some connections to the Common Core State Standards. Standards aren’t technology like computers and smartphones, but the CCSSM — particularly the Standards of Mathematical Practice — can be seen as tools for mathematical task design. There’s a lot in the world that could be mathematized, but having a set of standards can help make sure it’s done with the right content and practices in mind.
There was an interesting set of presenters and a jam-packed three-day schedule. I’m hoping to take some space here to summarize some of the presentations and moments that stood out to me while interweaving some of my thoughts, reactions, and questions.
Charles Fadel (Opening Address)
Michael Kaplan (Probabilistic Reasoning)
“The less we know about a subject, the more certain we are about our opinions. It’s the way our thrifty brains save our precious mental resources. But when we know what we are talking about, we willingly acknowledge complexity and uncertainty.”
“Too often we fail to engage students in the uncertainty and complexity of problems and, instead, we treat them as a receptacle.“
“Mathematics should be useful, we are told, but useful for whom?”
“We make a sad mistake when we propose that studying mathematics will lead to dream careers.”
“So much effort in repetition designed to get some idea in the mind of a student is time wasted.”
He said much more, but these were the quotes that caught my attention. Lots to think about here.
Jon Star (Cognition and Neuroscience)
Keith Devlin (Mathematical Thinking)
Sverker Lundin (The Drift Towards Purity)
- Be careful about higher goals in education (democracy, thinking, etc.).
- Think about how what students do in schools could lead to these goals. Expect no magic from the subject matter of mathematics.
- Open up other ways for students to approach mathematics than through tool-free pure problem solving
- We have inherited a problem and its solution. We should rethink both
Conrad Wolfram (Stop Teaching Calculating, Start Teaching Math)
Merrilea Mayo (What Math Do People Use in the Workplace)
- The correlation between education and career readiness and success is 0.1 (almost none).
- More than 90% of people will never use more than 6th grade level mathematics.
- In practice, even engineers were not using differential equations, the software was.
- She found that people were learning even their “applied math” outside of school.
- There is evidence to suggest that taking math does not mean you know or can apply it.
Bryan Meyer (Interdisciplinary Education)
- Who decides what counts as mathematical knowledge and mathematical activity? How do they decide it? What are the benefits and consequences of how we answer that question?
- Is it interdisciplinary education that we are after or is it anti/undisciplinary education? What are the benefits and consequences of defining a discipline of mathematics?
- Related to both above, should we operate under the goal of “mathematics for all” or “all people are mathematical?” How will our actions be affected by the place that we choose to operate from?
What Did I Learn?
- I have been thinking a lot about the reasons we hear/accept/give ourselves for why we teach math. Many of the common ones were subtly brought into question by a few speakers. I’m not going to give my thoughts on that here and now, but I think it is worth considering. I also think it is worth considering how our answer to that question might impact our work with students – for better or worse. When we push towards something, what are the effects of that push?
- The premise of the conference was “what should students learn in the 21st century?” That’s a big question and I just never felt settled on the idea that changing what students should learn will somehow change anything at all. I guess I’m left wondering and thinking about what it is we want to change in math education? Enjoyment? “Success?” Something else?