January 24, 2012

How Many Cookies?

I had the idea for this project though a slightly complicated sequence, but because each part is individually awesome I'm about to send you into a sentence overflowing with links.  Back in December I was listening to a Radio Lab podcast, where they featured Roman Mars who shared his podcast (99% Invisible) on Nicholas Felton.  Okay, I'll wait while you head to iTunes to subscribe to both podcasts...

Before students headed off for their 11 day winter break, I gave them this assignment (among others):

Nicholas Felton knows a lot about his life, because he keeps track of everything to write an annual report at the end of each year.  This year will be his seventh reporting on topics from how many types of nuts he ate (7) to types of transportation used (23) to iTunes songs played (16,682).  If you have a chance, check out Feltron.com for lots more statistics. 
We will be reviewing data analysis next semester, and vacation is a great time to collect data.  Ms. Cardone will be recording two things: # of cookies consumed and # of people spoken to.  It is up to you what you want to keep track of, but it needs to be some information that 1. You can keep track of with a number per day 2. You don’t mind sharing 3. Will be more interesting than all 0’s.
I was really excited going into vacation, because it was vacation (obviously), I would be eating lots of cookies and hardly talking to any people- peace and quiet!  I had no idea how many students would buy in to this part of their vacation packet, but I figured we could just share data if need be.  Upon return from break I was pleasantly surprised to discover that most students had actually tracked something!  Those that hadn't could usually come up with something from memory or looking in their phones and we had an awesome variety of topics.  The tallies ranged from sirens heard to video game stats to text messages to hugs.  

I collected all the data immediately and then a few weeks later I managed to get the laptops reserved to do our analysis.  My school is having a literacy initiative where students complete the writing process at least once a quarter in all their classes.  The prompt was "Analyze one aspect of your vacation using data" and we started 'brainstorming' by computing all the M's (mean, median, maximum etc.).  Then they began their paragraphs with a thesis statement (I spoke to relatively few people over vacation.) and supported it with evidence (Most days I only spoke to 3 people, my immediate family members).  The end results were a lot of fun to read.  It was much easier for students to analyze something they had experienced than random data; plus I got some insight into what their vacation and daily life experiences are.  Thanks to Radio Lab, 99% Invisible and Nicholas Felton for a great activity reviewing data analysis!

January 21, 2012

The Centers: Circumcenter, Orthocenter, Incenter and Centroid

My geometry classes recently finished a unit on finding 'the middle' of triangles.  The word middle was thrown out constantly and I regularly had to ask for clarification- what kind of middle?  the middle of what?  In fact, I think I might make that the last question on the test- "Locate the middle of this triangle and defend your answer."  Those should be fun to read (don't worry, I have no intention of tricking my students, anything goes so long as it's justified accurately).  Last year I approached this unit with some incredulity- I have to get them to memorize the definition of circumcenter??  I don't remember ever even seeing this word before!  But we all got through it just fine.  Since then I attended conferences at the Education Development Center in Newton and the Park City Math Institute where we not only used these fancy words, but also talked about just how cool their properties are.

This year I started out much more organized.  I wanted the students to have a clear reference sheet with all the information we learned and discovered in one place, so they started the unit with this chart:

Triangle Segments

It's nothing fancy, but since I introduce the different segments on different days, and then they continue discovering characteristics as we go, it was hugely helpful to be able to say "make sure to add this definition/observation to your chart." Whenever they had a question I could refer them to their chart; this became especially useful as we did the following two activities:

The airport problem is a fairly typical perpendicular bisector/circumcenter problem that I copied from my textbook.  There are two characteristics of this handout that I like which aren't typically included.  First, I ask students to check that the airport they found is actually equidistant from the cities, but not in a way that just says "check your answer."  By asking students to convert their distances to miles we hit on scale factors and map skills, while hiding the question "are you sure that point makes sense?"  The second thing is asking whether that point really makes sense.  I'm hoping for an analysis of highways or whether there's enough open space for an airport there.  Most of the time I'm all for problems without context, but if we're going to use a a context let's really analyze it!

Airport Investigation

Finally, we try making mobiles.  The main goal of this activity is to discover that the centroid is the center of mass, but it has the added benefit of showing some of the properties of equilateral and isosceles triangles.

Balancing Triangles

I think I did a much better job of emphasizing how amazing it is that all of these lines are always concurrent and of comparing/contrasting the segments and points.  It's amazing what a year's time can do to change my enthusiasm for and confidence in a topic.  Next year I might add a variation on missclacul8's fire station problem (but with a lot less scaffolding) since the concept of regions didn't appear in the unit at all this year.  Who know's what I'll be thinking in another year's time though!

January 18, 2012

My Student Self

Last week I started an online course in Geometry and Measurement run by the EDC (love them!) for my district and a neighboring one.  Being a student in a different environment has me reflecting on how I would react to being a student in my own class.

We started the course with a face-to-face meeting.  I'm not going to lie, I was a little resentful that I needed to attend since I was looking forward to taking the course from my couch at home on my own schedule.  The resentment only grew as I was instructed on how to sign in (which we'd been asked to do ahead of time), use a forum (something I've been doing for years) and use GeoGebra (I teach my students to use it).

When I'm teaching I need to make sure not to rehash ideas that students already know.  I hope that when we discuss homework students view it as an opportunity to check their answers, not just the class re-doing the work they already did.  I try not to go over every little step but I need to make sure that when the class gets it, we get on to the challenging problems.  Also, I need to explicitly ask students to be resources for each other when we study a topic some of them are familiar with.  I would have been happy to share what I knew about GeoGebra- but being asked to draw a triangle was just boring.

Each session of the course starts on Wednesday.  Last week I'd finished all the readings by Thursday night.  This week? I finished all the readings and activities by 8 pm, Wednesday.  I'm a nerd and am happy to own that label.  But I'm also really efficient; I'd rather just take a couple hours to go through all the material with complete focus and no distractions than drag it out over a week.  This is one of the reasons I signed up for an online course- it's very self-paced and allows me to be independent.

While there is opportunity for students to do their homework whenever they like within the 2-4 day gap between classes (block scheduling), there is very little room for self-pacing in the classes I teach.  Each problem set I assign in class is low-threshold/high-ceiling, but there is no room to move on to the next unit until the whole class is ready.  At this point it would be logistically a nightmare to allow students in geometry to move forward since we do so much experimentation and compiling of data to form conclusions.  My basic algebra class does spent part of their time on fact practice and that portion of the course is entirely self-paced.  I'd be interested to see if there was anything similar in geometry to allow students more control over at least part of the curriculum.

Finally, we are required to post in the discussion forums throughout the week.  That's the one piece I haven't done yet, and may no do until Friday.  I own my nerd status, but that doesn't mean I'm not hesitant to look like an overachiever.  This is especially true since I don't know half of the people in the course (they're from another district).  I will be an overachiever in the end (last week I posted more than twice as many comments as were required) but I won't totally stick my neck out there... yet.

It's really important that students feel comfortable in the classroom.  By this point in the year I forget that this means not just trusting the teacher (me) but also their peers.  I am shocked when a student is helping to return papers and doesn't know a classmate's name.  All of them have a few people they are okay working with (I've shuffled pairs until everyone has a few partners they gravitate to) but participating and asking questions in front of the whole class means they need to know that the entire class will be respectful and supportive of them.  And that includes not getting called a brown-noser (I really hate that term, it's just so gross).

I'm excited for the rest of this course as I continue to analyze my learning style, as well as the materials presented.  So far we've looked at essential knowledge for students to arrive to geometry with and methods for examining student work.  The technology components are coming up soon.  They include: an iPad to use for the semester (to be grouped into a set teachers can reserve next year), a new desktop to use at school and my new smart board!  I'd love to hear your suggestions for iPad apps and the smart board (applets, features, uses, anything- I've never used one before!).