Tag Archives: mathematics

Semester in Review: Combined IB SL/HL Mathematics Class

This past semester was tough. I've always taken on more than I probably should, but I hit my limit and need to change things around.

The biggest element of the challenge came from my combined IB mathematics standard-level/high-level class. It's a given that this combined SL/HL situation isn't ideal. My internet searches on ways this has already been done successfully haven't yielded much, aside from people saying that this is a bad idea. Another important given, however, is that I had input into the schedule building last year, and saw that our size and staff prevents this from being done any other way. The way I see it is as a design challenge: given that SL and HL students are working in the same room, what do I do to optimize that time?

My lesson planning process has been pretty consistent over the past few years. My learning standards for each group are based in the curriculum documents provided by IB. These are pretty solid documents in mathematics, and I don't tend to struggle there. Some standards are common to SL and HL, with HL specific additions added to my standards descriptions. The HL specific standards have also been pretty easy to parse out of the documents.

From the standards, I piece together pacing based on my experience and knowledge of my students. I've been teaching them all for the past two years at least, so I feel comfortable knowing how to push them. For each lesson, I curate a set of problems as the benchmarks, work out the prerequisite skills, and then figure out what students are doing at each stage of the class. At this point (which is usually about forty-five minutes in), I identify how direct instruction fits into the sequence, if it needs to be there.

This is where the separation between the groups gets tricky. I don't always have the SL students necessarily do the same warm up questions as the HL. The HL students might be given one basic problem, and another that forces them to figure out a need for a new method, find patterns, or attempt to generalize based on observations. While the HL students do this, I am debriefing with the SL students, giving them a mini lesson on the objectives of the day, and then setting them off to do some practice. This frees me to work with the HL students, give them a mini lesson on their objectives, and then get them working in a team. I circle back to the SL and work with them wherever needed.

I let the HL students flounder a lot when they are working together. That productive struggle leads to a need for me to come in and nudge them in the right direction with the right question or observation. In a perfect world, I don't need to nudge and the students figure it out themselves, but the dense reality of the curriculum doesn't allow for too much discovery.

This process, on the whole, is exhausting. It's only one of my classes to prepare on any given day, though the block schedule gives a lot more flexibility to do this than if I only had 45 minutes. Out of necessity, I can't spend time fixated on the perfect pivotal questions. While it is easier to do teacher centered instruction, the planning I used to do just isn't practical. I do a lot less instruction and a lot more throwing my students into problems and cleaning up issues along the way. Students wanting a clear set of instructions from me aren't getting them, which admittedly bugs me sometimes. In the long run, these students are spending more time figuring things out on their own and talking to each other, which makes me feel better about the situation. I just wish I was a better curator of materials to make this more smooth for students.

I have identified some ways I plan to change things around for second semester and lighten the load. The curation piece is the big one. Choosing good problems for each group to work on together is the most important element of that work. My direct instruction is then focused on leading students through the tough parts of the thinking process, and then getting out of the way to let them finish the job. The downside to this is that the completeness of my class notes decreases, but I'm not convinced students look back at those notes frequently anyway. There is a lot of good material available online to help students through the basic skills, and my time might be better spent finding and collecting that content for students to work through on their own.

I also feel the need to improve the quality of my interactions with each group. This is especially difficult when I am switching gears so quickly. Some SL or HL discussions don't fit neatly into a twenty minute interval together while the other group is working. I've decided that two blocks of every two week cycle (five blocks total) will be HL specific time. This means the SL students will have time to work on their own and help each other, and I can spend longer intervals of time working specifically with HL students on their exclusive content. The SL students undoubtedly have work to do for their other IB courses, and have expressed an interest in having time to work. The dedicated HL time will also mean the time spent with SL students and on common content becomes more streamlined and focused.

I'm always looking for ways to improve my workflow, so your suggestions are, as always, very welcome.

Simulations, Models, and the 2012 US Election

After the elections last night, I found I was looking back at Nate Silver's blog at the New York Times, Five Thirty Eight.

Here was his predicted electoral college map:

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...and here was what ended up happening (from CNN.com):

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I've spent some time reading through Nate Silver's methodology throughout the election season. It's detailed enough to get a good idea of how far he and his team  have gone to construct a good model for simulating the election results. There is plenty of description of how he has used available information to construct the models used to predict election results, and last night was an incredible validation of his model. His popular vote percentage for Romney was predicted to be 48.4%, with the actual at 48.3 %. Considering all of the variables associated with human emotion, the complex factors involved in individuals making their decisions on how to vote, the fact that the Five Thirty Eight model worked so well is a testament to what a really good model can do with large amounts of data.

My fear is that the post-election analysis of such a tool over emphasizes the hand-waving and black box nature of what simulation can do. I see this as a real opportunity for us to pick up real world analyses like these, share them with students, and use it as an opportunity to get them involved in understanding what goes into a good model. How is it constructed? How does it accommodate new information? There is a lot of really smart thinking that went into this, but it isn't necessarily beyond our students to at a minimum understand aspects of it. At its best, this is a chance to model something that is truly complex and see how good such a model can be.

I see this as another piece of evidence that computational thinking is a necessary skill for students to learn today. Seeing how to create a computational model of something in the real world, or minimally seeing it as an comprehensible process, gives them the power to understand how to ask and answer their own questions about the world. This is really interesting mathematics, and is just about the least contrived real world problem out there. It screams out to us to use it to get our students excited about what is possible with the tools we give them.

Why my trip to New Zealand will make me a better teacher this week....

I just returned today from an amazing three week tour of New Zealand with my wife. My plan is to post photos and captions somewhere in cyberspace, though I haven't figured out exactly where, and given the start of the new semester this coming week, it may take some time before I am able to do so.

Given that it was the end of the semester before we left, there was no need to even think of bringing work along. Instead, I was able to spend my time focused on the most breathtaking 3,500 kilometers of driving I've ever done, giving mountain biking a try (with the scars to show for it), and staring down trails like this:

It amazes me how taking time to completely take my mind off of work and teaching somehow tends to result in doing some of my best brainstorming about work and teaching. Making time for genuine renewal is a real productivity booster. I read The Way We're Working Isn't Working by Tony Schwartz a couple years ago towards the end of the school year, an excellent book which explores this idea in depth. I found myself agreeing with all of the concepts then, even though I had done the complete opposite throughout the year. It is counter-intuitive to take a break in the midst of stress - you think about how many little tasks you can get done in the ten minutes you might spend taking a walk, or the thirty minutes you might spend running a few miles, and it becomes too easy to rationalize not  taking a break even though there is plenty of evidence to show that it does good things for you.  It's the same principle behind the Google twenty percent rule through which employees are given 20% of their work week to work on whatever projects they want to work on.

I made the decision to keep most of my tech toys at home on this trip. I checked email occasionally and looked at tweets, but was otherwise fully immersed in the various adventures we had scheduled for ourselves. It was the right decision, including from a teaching standpoint for this reason: I find myself starting the semester with a big list of ideas for activities and potential projects to engage and involve students through my classroom. I am excited to share my vacation with students on a basic level, but am more excited to show how bug splatters lead to finding definite integrals, or how hiking on a glacier made me think about self similarity. I will share those ideas as I put some structure to them and share them with students over the next week or so.

In the meantime, here is just a taste of another #anyqs that is brewing at the moment:

Finally, a video look at this curious landmark from the North Island: