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.
]]>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.
]]>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.
Pose two lesson objectives. For instance:
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.
]]>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?
Resources
Raymond Johnson is a graduate student at CU-Boulder. He blogs at MathEd.net and tweets @MathEdNet.
]]>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.
Resources
Raymond Johnson is a graduate student at CU-Boulder. He blogs at MathEd.net and tweets @MathEdNet.
]]>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.
“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.
Resources
Bryan Meyer teaches at New Tech High in San Diego. He blogs at Doing Mathematics and tweets @doingmath.
When someone says “Yeahbut has a conference session ever actually caused you to change your practice?” I immediately point to Peg Smith. After catching her 5 Practices session two years ago at NCSM Indianapolis, strategies to facilitate productive classroom discussions replaced “Who wants to go next?”
No surprise I was looking forward to Dr. Smith’s NCTM Denver session. The description:
Although there is a growing consensus that the grades 7–12 curriculum needs greater emphasis on reasoning and proof, research shows that most textbooks offer limited opportunities to engage in these practices. We will focus on how to modify tasks to give students more opportunities to engage in reasoning and proving.
Smith began by contrasting modifying textbook tasks with Dan Meyer’s approach, which she described as rebuilding. Smith was highly complimentary of Dan’s work but the focus of her session would be on small, easy to implement changes; the problems themselves would remain similar to the original textbook versions.
Smith presented three tasks, each time giving us time to compare and talk about the before and after versions before she discussed them in depth.
Task A, making conjectures about the sum and product of any two odd numbers, received the least amount of modification. To my colleagues and me, adding “explain why” (with some language related to what it means to prove or disprove a conjecture) did move this task beyond fill in the blanks, but it just seemed like a given. We wondered about making other changes to the problem, such as removing the sentence stems, which are complete conjectures but for the last word.
The modifications to Task B, the geometric construction of a parallelogram, were more noticeable. In Task B’, three conjectures are possible whereas, in Task B, students are told the construction will produce a parallelogram. In Task B, reasoning and proof consists of stating a theorem found in the margins of their textbook. In Task B’, reasoning and proof consists of students themselves creating mathematical arguments. “Authority in math class should lie within the students and the math, not in the teacher or the text,” said Smith.
There were some significant improvements to the third task, a visual pattern involving trapezoids. In Task C’, students are first asked what they notice about the visual pattern. This is significant because these observations can be used later by students to prove their generalization. In Task C, the visual pattern serves no purpose other than to generate the sequence 5, 8, 11, 14 for students. Smith asked us to note the placement of the question about the perimeter of the 12th figure in each version of this task. In the textbook task, it follows the question about the formula; it serves to verify that the formula is correct. In the modified task, it comes before the question about the generalization; it encourages students to generalize.
Smith ended her session by asking “Do these modifications make a difference?” I think so. These modified tasks do give students more opportunities to engage in reasoning and proof. No question. However, I couldn’t help leave the session wondering if, by now, these modified tasks shouldn’t be our starting points.
Resources:
Chris Hunter blogs at Reflections in the Why and tweets @ChrisHunter36.
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