I have to confess that I have not always been comfortable teaching probability. I’m fairly certain that I didn’t learn anything about it in high school (NYS Regents mathematics way back in the day BEFORE Sequential I, II and III), and I never took a course which included it in college. My knowledge is self-acquired and taught – through preparing for certification exams, Praxis exams, and teaching. In the 5 years I’ve been teaching the topic, I’ve worked and reworked my lessons in Algebra 2 to bring my own understanding to a conceptual level which is deep enough to communicate to my students, and have relished the loving addiction to Pascal’s Triangle I have thus created in many of them.
This term in Discrete Math, I took the plunge and taught a unit (using lessons generously shared with me by a colleague at school) based on games of chance – dice, spinners, flipped quarters, etc. After two weeks of playing and analyzing games, we spent several days learning about Expected Value, relatively simple to calculate (in our examples) but tricky to understand. Once I was convinced that the students knew how to approach an expected value problem, we worked through the famous Money Duck lesson, developed by Dan Meyer.
Attendance is an issue in my Discrete Math classes, so over the course of the lesson and the task that followed (a total of three and half days), I replayed the video for students who were either absent physically or perhaps mentally. I never anticipated how much 16 year old students would enjoy watching Dan Meyer wash his hands repeatedly. The class came to a standstill every time I turned it on. This enthusiasm for the video carried over to their analysis of the probability distributions.
My colleague designed an extension for the lesson, similar to Dan Anderson’s activity. The students designed their own “Money Animals”, complete with a price, distribution, and an expected value. This was all done on one sheet; the design, price and distribution were visible to all, while the calculations were on the back. After everyone had finished, we had our Money Animal Bonanza.
The students were allowed to purchase two Money Animals, based on the desirability of the product itself and its distribution.
At first, just a few students wandered over to the Money Animal Mart. I was impressed with how thoughtfully these early shoppers examined their options. And then EVERYONE rushed the board, eager to debate the pros and cons of design and potential gain. The results were interesting, and in some ways predictable.
(1) There’s an artist in every class, waiting to draw a gorilla.
(2) Everyone loves unicorns.
(3) Marketing is everything, and frequently outweighs sensible decision-making.
The clock ran out, and I announced the ‘sales’ results and the concomitant expected values. The potential earnings on the BAPE soap were pretty low with its steep price of $35, but the students unanimously supported its cachet.
I would like to have the students complete some sort of analysis which would make the relationship between the price, the distribution and expected value more concrete. Maybe expand the Money Animal Mart idea, give them Monopoly money to shop with, and find a way to calculate everyone’s profit as a designer, and winnings as a shopper? I’m looking forward to running this activity again with further refinement – because if I can get the students up, moving and this interested, there’s definitely a bigger future for this.
By the way, Happy Halloween!
On a break just 8 days into the Spring term (ironic as that denotation may be), I’m feeling more energized than my 7 a.m. start time would suggest. It’s a great relief after an angst-ridden fall term, and while I am not looking this gift horse too closely in the mouth, I am reflecting on how I managed to scale the wall that felt insurmountable just a few weeks ago.
In Algebra 2, the term begins with an introduction to Trigonometry, which makes me unspeakably happy. We started out by discovering radians with paper plates, exploring arc length and special right triangles (I am not sure why they are so special, Dan Meyer, but the universality of those ratios resonating throughout math and design is, in some literal way, awesome. Call me crazy, or nerdy, or both.)
Proving the Pythagorean Identities was also a wondrous exercise, even eliciting applause from a student who clearly has a future as a math teacher. I’ve got a better understanding of how to sequence the content this year while keeping pace with my department’s calendar, and I’m finding time to infuse class with discovery. Thanks to the generous assistance of Audrey McLaren and the thoroughly spot on webinar by Crystal Kirch, I’ve begun some forays into the flipped classroom. I started with a VoiceThread on reviewing the basic trigonometric functions, which met with a lot of student approval and enthusiasm. I wish there were a few more hours in the day to incorporate all the ideas I’ve got, but I’m committed to starting to build my own library of flipping resources. More to come.
We’ve also gotten off to a great start in Geometry, due to several factors. The programming office shuffled the students between the sections of the course, and the resulting rosters are more balanced, with some of the more toxic behavioral combinations disassembled. There has been a 4th section of the class created – I was teaching three of them in the fall – and as a result, my BFF at work and I are planning together; he has been given one of the sections to teach solo, and we are co-teaching the ICT class (never mind that neither of us is a special educator – that’s a long story, and another blog post). This is the first time in a long while that I have had the opportunity to engage in true common planning with a like-minded colleague, and it has made a huge difference in alleviating the stress and isolation involved in creating a new course single-handedly. Mr. P and I have always shared ideas and experiences, but as c0-teachers, there is a true collaboration happening, which fosters more thoughtful planning. In trying to be always on the same page in a busy classroom (aka the 3-ringed circus of math), we have debated classroom decisions, pushing back on each other’s thinking, and in the process, crafting more authentically reflective policies and procedures.
It was gratifying to see that the students who had been in the class last term, fell quickly back into the established routines of the Daily Quiz*, the Interactive Notebooks, and collaborative work at the tables. Bringing the new students up to speed on the Interactive Notebooks has been more of a challenge; we spent a lot of time setting them up and working on the intent of the notebooks in the fall. Again, the group at each table provides a support for the newbies.
We spent the first two weeks reviewing special quadrilaterals, completing a graphic organizer (link below), a chart in which the properties of the polygons were compared and sorted in a Venn diagram, and Lisa Bejarano’s Always, Sometimes, Never activity. When we return from break, we will begin working on equations of lines as a lead-in to Coordinate Geometry.
When I go so long between posts, there’s always too much to say – some very, very dear friends of mine are relocating – one to California for graduate school, another to Shanghai for an amazing career opportunity. This has, inevitably, got me reflecting and rethinking choices I’ve made, and continue to make. But my own children continue to pursue their own unique interests and education with passion and talent, reminding me that every child deserves that chance – and brings me back, once again, to why I teach.
Speaking of my amazing children, which I can’t help myself from doing, my younger one is involved in a project to produce animated films in collaboration with NASA scientists working on the Fermi telescope – how completely cool is that? Read about it here, on the Tumblr run by Geo.
*The Daily Quiz is a low stakes formative assessment used as a warm-up for class which sparked an interesting twitter conversation last night, and which I may write a separate post about later this week.
This lovely wall hanging is posted high on the wall of the Exeter dining hall. It looks like a quilt, but is actually composed of painted wood. This evening, I ate dinner beneath it, sitting at a table with a group of teachers, including the first person I met at the conference. When I said, “It’s so good to see you – I haven’t seen you since the conference began!”, she replied, “Yes, that was yesterday morning.” The days have been so jampacked with ideas, conversation and laughter, I feel as if I have been here for a week already – in a very good way.
I managed to put some homework problems on the board, and explain my answers during my “Hits of Higher Math” class, and even ask a few questions. We moved from Real Analysis to Topology, which I find somewhat more accessible as we have [temporarily] put aside discussions of infinity. I joined forces with another reticent student (he is actually a Political Science person, recently conscripted to teach a math class at his school in Arizona) and we formed a mini-study group, and worked on the homework together today. In the iPad class, we were given a demo of an amazing product called Fluid Math (http://www.fluiditysoftware.com/), which allows you to hand write equations on a tablet, transform them into graphs, and create sliders with which you can manipulate graphs. This is a very bare bones description of what this app-suite can do – it is a tool that will definitely take time for me to play with, and imagine some of its applications in my classroom.
But the big bonus of the day was the evening speaker, Bruce Dixon, founder of the Anytime Anywhere Learning Foundation (http://aalf.org/), who gave an energizing and thought-provoking lecture on Reimagining School with Technology. Beginning with Ron Amara’s quote, “We tend to overestimate the effect of a technology in the short run and underestimate the effect in the long run,” Dixon looked at the current uses (or lack thereof) of technology in the classroom, the amazing communicative and learning power that technology provides for our students, and the shifting paradigms in economics and society that constant access to information and each other creates. He posed provocative questions about the current unimaginative use of technology in many schools, and the restrictions we place on internet access in schools which might be impeding educational progress. He suggested that we need to ENCOURAGE our students to talk to strangers, globally, to gather information from appropriate sources, and that we need to not just plan for technology implementation in our classrooms, but REIMAGINE the role that technology will play in education. When the talk began, I thought I understood how I wanted to use technology – now, I can barely write about it – not because I am confused, but because the idea is so BIG, and so IMPORTANT.
Dixon is an incredibly dynamic speaker – he spoke for over an hour without notes. He did have a presentation to which he referred, but he had a room full of 200 teachers hanging on his every word. I had the opportunity to speak with him afterwards, and found him to be incredibly optimistic about the future, despite the actual title of his lecture which was “Is it possible that we are seeing the END OF SCHOOL as we know it?” He believes in the power and talent of our children and students, and also in the power of the participation of educators in this process. We discussed energy and water shortage, Van Hiele levels, making fires with bow drills, and Dan Meyer’s Pyramid of Pennies task (http://threeacts.mrmeyer.com/pyramidofpennies/) – luck us to hear Dan Meyer speak on Wednesday!