Exactly where [a lesson] moves depends on such complex factors as the structures of those present, the context, and what has been anticipated. It may move toward more formulated understandings, if such formulation is relevant to the play space or if it becomes part of a further exploration. It may simply move to other sorts of activities. This, of course, is not to say that we should just allow whatever might happen to happen, thus abandoning our responsibilities as teachers. Rather, it is to say that we cannot make others think the way we think or know what we know, but we can create those openings where we can interactively and jointly move toward deeper understandings of a shared situation.
(Davis, 1996, p. 238-39)
My Grade 9 students are currently working on recognizing, analyzing, graphing, and solving problems involving linear relations. Linear relations lend themselves so naturally to describing patterns à la www.visualpatterns.org, and this is precisely how we got our toes wet in the topic: For several days, my students had been analyzing, extending, and (productively) arguing about a variety of linear and non-linear patterns. The intention of these first few lessons was to have students develop (or, in some cases, refine) an understanding of constant and non-constant change and to connect patterns in pictures to patterns in tables of values.
“What you do speaks so loud that I cannot hear what you say.” ― Ralph Waldo Emerson
Since the first day of school, I’ve been working hard to try to establish a classroom culture where students feel comfortable taking risks, asking questions, sharing and building on each others’ fully-formed and partial ideas, and acknowledging and correcting their mistakes; where all students feel that their contributions and questions are valuable and worthy of consideration. I have tried to do so by pointing out (less often than I should) when a student or a group is exemplifying one of these norms, by waiting (again, less often than I should) after questions and contributions to give more students the time they need formulate and share their ideas, by giving tasks that are accessible to a wide range of students and can be tackled with a variety of strategies, by eliciting and celebrating different solution paths, by highlighting different kinds of mathematical smartness (h/t Ilana Horn)…
In our division, classes on the first day of school are only 15 minutes long. By the time students settle in and introductions are made, there is hardly enough time to wrestle and play with an interesting math problem. I saved that for the second day. Instead of going through the syllabus, however, I gave the time over to my students to reflect on the following questions:
Last week, Dan Meyer wrote a brief reflection on Ed Beagle’s First and Second Laws of Mathematics Education:
The validity of an idea about mathematics education and the plausibility of that idea are uncorrelated.
Mathematics education is much more complicated than you expected even though you expected it to be more complicated than you expected.
The second law particularly resonated with me, a soon-to-be teacher. The more I learn about mathematics education, the more I realize that there is still so much to learn, and that anyone who says it’s simple is selling you something (Dan Meyer). My to-read list is growing longer and longer, even as I realize more and more fully that what matters most is not what I read, but what I do at the ground level with my students. (Side note: Last week I also began my foray into John Mason’s work – thanks, Danny Brown.)
The other day, I presented some students with the following game:
Last cookie is a game for two players. In this game, a number of cookies are divided between two cookie jars, and each jar has at least one cookie. Each player can take cookie in one of two ways:
By taking any number they like from just one jar or
By taking the same amount from both jar.
The winner is the person who takes the last cookie(s).
Some of you may recognize this as Wythoff’s game; on the NRICH website, which has a great online version of the game, it’s called Last Biscuit (I modified the name for a Canadian audience). If you aren’t familiar with the problem, I highly encourage you to give it a go – very easy to understand and play, but the optimal strategy is not particularly obvious.
I assigned this game as homeplay (get it?? because it’s a game… terrible, I know). Then, the other day, we picked it up again after about a week of rumination.
Last week, some incredibly talented students at our high school put on an evening of one-act plays. I was particularly excited to see A Charlie Brown Christmas, and I was not disappointed – I was so impressed by how well the kids brought the classic cartoon to life. However, the play that really gave me some food for thought that night was A Straight Skinny: a story about a high school algebra class that had been caught cheating on a midterm exam.
First things first, Pi Club has a new member – and SHE is a wonderful addition to the group! WOOP WOOP, girl power!
Today, we worked on two problems. First, the lightbulb problem (found here):
There are 100 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently switched off. The room has an entry door and an exit door. There are 100 people lined up outside the entry door. Each bulb is numbered consecutively from 1 to 100. So is each person.
Person No. 1 enters the room, switches on every bulb, and exits. Person No. 2 enters and flips the switch on every second bulb (turning off bulbs 2, 4, 6…). Person No. 3 enters and flips the switch on every third bulb (changing the state on bulbs 3, 6, 9…). This continues until all 100 people have passed through the room.
What is the final state of bulb No. 64? And how many of the light bulbs are illuminated after the 100th person has passed through the room?
They didn’t take too long to work this one out. Great conversations were had.