Wednesday, September 28, 2011

Wearing Pants

In math, we...
  • use letters at the end of the alphabet to denote variables, at the beginning of the alphabet to denote constants, and in the middle of the alphabet to denote whole number increments.
  • call 1/2 simplified, 2/4 unsimplified, and 4/3 improper.
  • multiply before adding.
  • avoid fractions like $\frac{2.5}{3\frac{1}{2}}$
  • use x * and ()() to mean the same thing
  • use x to represent the horizontal axis and y to represent the vertical axis
  • drop the multiplication symbol when multiplying numbers by variables
  • write 2N instead of N2
and so on and so forth.

It's important to remember that these are all conventions, not fundamental axioms handed down on stone tablets by the gods of mathematics.

Sort of like wearing pants. The world wouldn't come to an end if you stopped wearing pants tomorrow, but you might get some funny looks, you might not be taken seriously, and it might be more difficult to communicate with others.

Monday, September 5, 2011

Mathematics of Moving

I'm moving in two weeks.  Before I'm deluged with offers to help, I should warn you that spots for this coveted job are limited and will be determined by a random drawing weighted by how much you can bench press.  I know, I know, what am I doing moving two weeks into the school year?  Well, I'll leave that riveting story of greed, mystery, and real estate markets for another time.  Anyway, when looking for new apartments I have found a couple of websites indispensable:
  • Craigslist (in addition to finding potential places, also helpful if you're looking to buy a pile of dirt)
  • Padmapper (for seeing where that "prime location" apartment actually is)
  • Walk Score (a website that gives an address a score based on its walkability to places people like to go)
One of the coolest parts of Walk Score is that the metric for how these scores are determined is right there on the site.  Go ahead, check them out yourself.

Lots of math in here that could (if students were interested) spark great conversation.  What did I focus on?


So let's talk algorithms.  If you've read anything I've ever written, or if you've ever been stuck in a pedagogy conversation with me (sorry), you know that I strongly believe (more than Cher believes in life after love) that algorithms are a lot more meaningful and powerful when developed by students versus spewed by teachers. I know, when I put it in such neutral terms it seems obvious. Anyway, nothing new there. I've had kids develop their own algorithms for multiplying double-digit numbers, finding fractions between any two fractions, rewriting rationals as Egyptian Fractions, summing infinite geometric series, etc. What I haven't done, though, is expand my definition of algorithm beyond the procedural.

How's this for a project?
  • Pick something you care about that you'd like to measure on some scale (like how walking friendly your new digs are). Pick something that doesn't already exist.
  • Create a metric/algorithm to score this thing you care about. Other than being able to defend your metric and not end up with radically wonky results, the sky's the limit in how you do this.
I sort of threw this together and may wake up in the morning, reread this, and delete the whole thing writing it off as too crazy and too open ended. Even if this does happen, I may still try it out later this year in one or more of my classes.

Oh, and by the way, the Walk Score of my new place?  100 out of 100. :)

Sunday, September 4, 2011

Rethinking Lesson Planning

Every lesson plan template I've ever seen looks similar to the following:
Taken from:
While teachers use these templates less and less as they become more experienced (unless, of course,  they are faced with district mandates/evaluations/"we don't think you work hard enough so here's something else to do").  That said, I find that the paradigm of developing lessons in a linear fashion remains. I'm going to reach my goals by doing X, then Y, then Z. Issues I have with tempates aside, I have been recently thinking about alternative ways to conceptualize lesson planning (like any scaffold, a template can be helpful when starting but can also be limiting. I still rue the fact that in high school I was taught how to write 5 paragraph essays REALLY well, but was never given the freedom to break from this structure).

Here's my initial stab at a way to think about lesson planning that is less linear and, hopefully, more supportive of best practices around student exploration and constructivist learning. For now, I'm calling it a lesson web.

Created with Mindjet MindManager

At the center of this web is a central skill, concept, or habit. Some of your goals for the lesson (which you can write down separately) will be directly connected to this central topic. From this central topic, you can brainstorm three things:
  • problems 
  • potential methods for solving problems within the realm of the central topic
  • potential misconceptions related to the central topic
Relationship links (the red arrows) connect specific problems to specific methods and misconceptions. Building these relationships is crucial, as it will serve as a check to make sure you are giving students problems that address potential misconceptions and desired method (while I personally don't think teaching specific algorithms is necessary, I understand that being familiar with a standard algorithm can sometimes make communication more efficient).

At the next level, problems branch into possible variations, extensions, and generalizations.  In my classroom these are developed by both me and my students.  These problems can also be connected to a new central topic, making this not only a template for lesson planning, but really a curriculum map.

Misconceptions can also link to problems that will help expose or eliminate those particular misconceptions.

Things I like
  • When giving students the flexibility to create their own problems and develop their own methods, the teacher needs to anticipate what students might do. This structure supports this. Teachers can even use this to outline the order in which students will share out (for example, starting with students who used method 1, then method 2, etc).
  • This structure is easy to adapt. After the lesson, it's easy to add new methods students came up with that you didn't think about or misconceptions that students had. This also gives the teacher a tool to use the problems students create as a springboard for future topics (making new connections between problems and topics).
  • While I have the topic playing the central role, you could just as easily start with a problem or a misconception and build the web from there.
Things I don't like
  • There's no assessment (formative or summative) built into this. This is something that a teacher will have to think about in parallel to building their web.
  • Sometimes my primary goal for a class might lead to a web where one or more of these fields don't naturally fit in.  For example, if my goal is to promote communication between partners, I can imagine problems that might help meet this goal, but I'm not sure what a "misconception" would look like or what different "methods" students might develop. 
  • This is a lot more work than just going with my gut and adapting on my feet. :)
Next step? Developing a lesson web for a specific topic.


Friday, September 2, 2011

Return of the Jedi

I am surely overstating my qualifications, but I am back after a relatively long hiatus from posting. Hopefully there are still two or three people out there reading this.

This past summer was the first since I began teaching 12 years ago where I truly embraced the vacation part of summer vacation. Eight weeks of no summer job, no professional development, no classes, no graduate school courses or writing, and no blog writing (or even reading). I spent seven of those eight weeks traveling around the southwest, with a side trip to Maui for non-camping relaxation, and a very short jaunt to Florida to watch the last group to go into space from American soil in quite some time.

One of my pics from Antelope Canyon
Some highlights?
  • Visited 9 states and 13 National Parks. 
  • Won the license plate game! If you’re not familiar, this means we saw license plates of all 50 states. Also saw license plates from 6 Canadian provinces (I’m disappointed in you, Manitoba) and 2 Mexican states. 
  • 1 near death experience with a rattlesnake. 

I can’t say I didn’t think about math, school, and all things professional. Many of the things I thought about are drafts of what may turn out to be a hurricane of blog posts in the next week or so. One recurring theme was summer vacation itself (I know, meta). I don’t know in what part this is due to the political atmosphere and in what part this is due to the fact that this was my first real summer vacation, but I found myself hesitant to talk with strangers about what I do for a living and what I was doing this summer. Feelings of guilt, indolence, and “being judged for not having a real job” arose.

Roger ClemensWhile I intellectually understand that much of the vitriol towards teachers in the past year has nothing to do with teachers and education, it’s still hard to hear pundits and “reporters” spouting untruths about lazy teachers working 6 hour days for 8 months a year, making extravagant salaries and benefits. Deciding that a teacher’s work day begins and ends at the class bell is like saying that an NFL player only works 16 days a year, or that Jon Stewart works a half hour a day. And movie stars? Three movies a year, 2 ½ hours per movie; they’re working less hours in a year than real 9-5 workers work in a day. As for our extravagant salaries, in 2007 the Yankee’s pitcher Roger Clemens made more than $7,000 PER PITCH. But please excuse me for just picking on high profile/famous jobs. Farmers only really work in April and October when they’re planting and harvesting, right? Sure sounds nice…