This post comes to you at 6:22 AM on the first very cold and blustery day of the season. It’s the start of the short pre- Thanksgiving week, and I am looking forward to the 4 day weekend probably as much as my students. The harsh chill wind feels appropriate after the morning news; stories which contrast some Jewish support for Donald Trump with his anti-Muslim rhetoric and views are particularly upsetting this morning. After a lifetime of holding up the Holocaust and saying, “never forget, never again,” it appears that some of my cultural compatriots are doing exactly that. The fear my Muslim students expressed to me on November 9 stays with me, and I am wondering how I can make them feel safe, at least in my classroom. I’ve spent a lot of time thinking about privilege the last few weeks, and I can thank the upset in the election for this – perhaps the only benefit I can see right now.
On the agenda today: In my three sections of Algebra 2, the students will be working in groups on tomorrow’s exam. This is the first time I have tried this strategy, but, bolstered by input from Jonathan Claydon, Amy Hogan and my office mate, I’m hoping the communal efforts will boost student understanding of the content, and their independent demonstrations of mastery. My concerns include making sure no exams (or photos of them) leave the classroom, and students not making good use of their time together. In Discrete Math, we will beginning our unit on Problem Solving strategies, which is a distillation of the course I taught last fall. I will still be using problem sets from Crossing the River with Dogs, but I’ve come up with several different versions of each set to use for assessment. The summative project in this unit will involve the students creating problem sets of their own; again, I am trying to counter any inclination to over-collaborate (how’s that for a euphemism?).
It’s not even 7:30 AM and I volunteered to go on the spring trip to Quebec with the foreign language department. I don’t speak any French, but I’d love to visit Canada, and maybe they need a math teacher!? The sound of the wind is a howl in my office, which is located on a corner of the school building on the top floor. Here we go, Monday morning.
Two sections of Algebra 2 worked on the ‘practice exam’; many students commented that they found it a helpful exercise. From my view, the group review surfaced the topics that need the most study, and I was able to reiterate these areas to the entire class. For me, it was an opportunity to observe, deflect questions and refer the students back to one another for support. At the end of each class, the students were puzzled that there would be no answer key provided for this review, and that they needed to leave the papers with me. But I provided a review and practice sheet for them last week, complete with an answer key, as well as an assignment on deltamath.com with many practice questions. I think it may have dawned on some of the students that they were looking at the actual exam, and this will be the only time I can use this element of surprise. Hopefully, I will see better results and more work that evidences understanding tomorrow.
A new basketball league was formed in which each of the teams will play three games against each of the other teams. There are seven teams: the Antelopes, the Bears, the Cubs, the Dusters, the Eagles, the Foxes, and the Goats. How many games will be played in all?
The range of approaches was impressive, although very few students attempted to draw a picture for a solution. I saw charts, lists, tree diagrams, and on some papers, a simple but erroneous 7 x 3 = 21. Many students who realized that the Antelopes needed to play 18 games assumed that each of the other 6 teams would play 18 different games as well. But in each class, there was at least one student who understood that the number of games each team would play when calculated this way was double the actual amount. It was a clear learning moment for those students who had made the error – I hope. (Come to think of it, the student work on this problem would make good fodder for mathmistakes.org!) I drew a network sketch on the board to show how I calculated the answer, but it looked complicated to many of the students – I’m not sure I disagreed.
We moved on to Model Train Set:
1) Teachers make a lot of decisions throughout the day. Sometimes we make so many it feels overwhelming. When you think about today, what is a decision/teacher move you made that you are proud of? What is one you are worried wasn’t ideal?
I was very proud of my deflecting all student questions during the exam review today. I redirected the children back to each other, and answered their questions with more questions. And I think I managed to keep them from being furious with me while I was doing it.
Conversely, I think I could have pushed my Discrete Math students with some questioning a little more during the problem solving activity. I’m going to work on that in the lessons to come.
2) Every person’s life is full of highs and lows. Share with us some of what that is like for a teacher. What are you looking forward to? What has been a challenge for you lately?
3) We are reminded constantly of how relational teaching is. As teachers we work to build relationships with our coworkers and students. Describe a relational moment you had with someone recently.
I am feeling more confident in the relationships I am developing with people at Math for America. I’ve come a long way to get there, but that’s another story for another post (maybe).
4) Teachers are always working on improving, and often have specific goals for things to work on throughout a year. What have you been doing to work toward your goal? How do you feel you are doing?
I had four students come see me for extra help today in preparation for tomorrow’s exam – they came bustling in with their snacks in between classes and the school basketball game. They asked questions, helped each other, and worked away. I love when the kids are that comfortable in my office, and it lets me know I am creating safe spaces for them in which to be themselves.
5) What else happened this month that you would like to share?
Saturday night was Nerd Prom aka the Math for America Fall Function, complete with aerial entertainment, decagonal menus, and a mayoral speech. I said in my last post that November 2016 has not been my favorite month ever, but Saturday evening helped. Thanks, Math for America!
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.
I am a NYC Department of Education Common Core Fellow. This lofty title means that I have been trained (by the NYCDOE) in the evaluation of materials for alignment to the new standards, for ‘focus, coherence and rigor’, and for accessibility to diverse learners. Since my return from Exeter, I have been engaged in workshops in which we (my fellow Fellows and I) are reviewing and revising curriculum materials that have been created by school-based teams, some of which we headed. One focus of these reviews is the culminating task. It is important that the culminating tasks in each unit are structured so that successful completion of said task is clear evidence of mastery of those standards to which it is aligned.
As it turns out, despite our intentions to elicit independent demonstration of learning from our students, we, as teachers, seem to be almost unable – or afraid, more likely – to allow our students to engage in the ‘productive struggle’ in these tasks. “Too much prompting” is a frequent criticism, and “the task should be less teacher-guided.” The gap seems to lie between what we know our students OUGHT to be able to do, and what our actual classroom experience of their ability is. How do we make that shift ourselves?
In my view, the issue is less whether the Common Core standards represent the true direction in which education should move, but whether we can consistently raise our expectations of our students and of ourselves, ask questions that are truly open-ended and craft classroom experiences that allow our students to explore those questions – independently, cooperatively, collaboratively. I would love to be that teacher, and every September, I try again.
Of course, juxtaposed with these lofty goals are the new performance evaluation systems for teachers which include, as one criterion, student performance on standardized tests. It is not to difficult to understand why so many teachers will not leave student performance open-ended. But ultimately I think we need to trust ourselves and our students in order to harness the enormous potential in 21st century learning. Pie in the sky? Perhaps. But I didn’t become a career-changing teacher in my mid-40’s because I lacked imagination.